Lesson Addition, Subtraction and Multiplication by a number of vectors in a coordinate plane

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<H2>Addition, subtraction and multiplication by a number of vectors in a coordinate plane</H2>

In this lesson you will learn how to add, &nbsp;subtract and multiply vectors by numbers using component form of vectors in a coordinate plane.

The lesson is the continuation of the lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>&nbsp; and &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-coordinate-plane.lesson>Vectors in a coordinate plane</A>&nbsp; under the current topic in this site.

Recall that in a coordinate plane each vector &nbsp;u&nbsp; can be presented in the component form &nbsp;u = (a,b), &nbsp;where the real numbers &nbsp;a&nbsp; and &nbsp;b&nbsp; are the projections of the vector &nbsp;u 
to the coordinate axes &nbsp;x&nbsp; and &nbsp;y&nbsp; respectively &nbsp;(see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-coordinate-plane.lesson>Vectors in a coordinate plane</A>).

<H3>Adding vectors in a coordinate plane</H3>

Let &nbsp;PQ = u = (a,b) &nbsp;and &nbsp;QR = v = (c,d) &nbsp;be the two vectors in the coordinate plane &nbsp;(Figure 1).

P = (x1,y1) &nbsp; and &nbsp;Q = (x2,y2) &nbsp;are the initial point and the terminal point of the vector &nbsp;u. &nbsp;Real numbers &nbsp;a = x2-x1&nbsp; and &nbsp;b = y2-y1&nbsp; are &nbsp;x- &nbsp;and &nbsp;y- projections of the vector &nbsp;PQ.

Q &nbsp; and &nbsp;R = (x3,y3) &nbsp;are the initial point and the terminal point of the vector &nbsp;v. &nbsp;Real numbers &nbsp;c = x3-x2&nbsp; and &nbsp;d = y3-y2&nbsp; are &nbsp;x- &nbsp;and &nbsp;y- projections of the vector &nbsp;QR.

We are going to calculate the sum &nbsp;u+v&nbsp; of these vectors.

The procedure of adding vectors in a plane is described in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A> &nbsp;(the &nbsp;triangle rule&nbsp; or the &nbsp;parallelogram rule).

We suppose that the terminal point of the vector &nbsp;u&nbsp; coincides with the initial point of the vector &nbsp;v &nbsp;as it is shown in &nbsp;Figure 1. &nbsp;If it is not so, &nbsp;we can make the parallel translation of the vector &nbsp;v 
to superpose its initial point onto the terminal point of the vector &nbsp;u, &nbsp;according to the procedure adding vectors of the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>.

<TABLE>
 <TR>
 <TD>
{{{drawing( 282, 262, -2.0, 12.1, -2.0, 11.0,
 grid(1),
 line( 2.0, 2.0, 5.0, 7.0), 
 locate( 1.8, 1.9, P),
 locate( 3.2, 5.9, u),

line( 5.0, 7.0, 4.6,  6.6),
            line( 5.0, 7.0, 4.85, 6.5),

line( 5.0, 7.0, 11.0, 9.0),
          locate( 4.5, 7.8, Q),
          locate( 7.3, 8.8, v),

line( 11.0, 9.0,  10.5, 8.7),
            line( 11.0, 9.0,  10.5, 9.02),

red(line( 2.0,  2.0, 11.0, 9.0)),
            line( 11.0, 9.0, 10.6, 8.5),
          locate( 11.1, 9.8, R),
          locate( 7.3, 6.0, u+v),

green(line(2, 0, 4.8, 0)),
          locate( 3.2, 0.9, a),

blue(line(5.2, 0, 11.0, 0)),
          locate( 7.9, 0.9, c),

green(line(0, 2,  0,   6.8)),
          locate(-1.4,  4.8, b),

blue(line(0, 7.2, 0, 9)),
          locate(-1.4, 8.3, d),

locate( 1.8, 0.9, x1),
          locate( 4.8, 0.9, x2),
          locate(10.7, 0.9, x3),

locate( 0.3, 2.4, y1),
          locate( 0.3, 7.4, y2),
          locate( 0.3, 9.4, y3)
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. &nbsp;Adding vectors
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in a coordinate plane (1)&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE>

According to the &nbsp;triangle rule&nbsp; of adding vectors, &nbsp;the sum of the vectors &nbsp;PQ&nbsp; and &nbsp;QR&nbsp; is the third side &nbsp;PR&nbsp; of the triangle &nbsp;PQR&nbsp; with the initial point &nbsp;P&nbsp; and the terminal point &nbsp;R. &nbsp;The &nbsp;x-projection of the vector &nbsp;PR&nbsp; is &nbsp;x3-x1. &nbsp;It is clear that &nbsp;x3-x1 = (x3-x2) + (x2-x1). &nbsp;In other words, &nbsp;the x-projection of the vector &nbsp;QR&nbsp; is equal to &nbsp;a+c.

Similarly, the &nbsp;y-projection of the vector &nbsp;PR&nbsp; is &nbsp;y3-y1. &nbsp;Again, &nbsp;it is clear that &nbsp;y3-y1 = (y3-y2) + (y2-y1). &nbsp;In other words, &nbsp;the y-projection of the vector &nbsp;QR&nbsp; is equal to &nbsp;b+d.

Thus the vector &nbsp;u+v&nbsp; has the &nbsp;x-projection equal to &nbsp;a+c &nbsp;and the &nbsp;y-projection equal to &nbsp;b+d. &nbsp;In the shortened component form, &nbsp;this means that &nbsp;(a,b) + (c,d) = (a+c,b+d).

This is the formula for adding vectors in a coordinate plane:<BLOCKQUOTE><TABLE BORDER=2>
 <TR>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;(a,b) + (c,d) = (a+c,b+d)&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE></BLOCKQUOTE>One can derive this rule geometrically, too, by playing with the projection segments in &nbsp;x- and &nbsp;y- axes that are shown in the &nbsp;Figure 1&nbsp; in green and blue colors. &nbsp;The geometric proof is based on the fact that the co-named projection segments of the vectors &nbsp;u&nbsp; and &nbsp;v&nbsp; join each other.&nbsp; Such a proof should consider different dispositions of the vectors &nbsp;u&nbsp; and &nbsp;v&nbsp; that create positive and negatives values of the projections. &nbsp;The algebraic proof provided above takes into account all these details automatically.

Example 1. &nbsp;Find the sum of the vectors &nbsp;(3,5)&nbsp; and &nbsp;(6,2)&nbsp; in the coordinate plane.

Solution. &nbsp;We have &nbsp;(3,5) + (6,2) = (3+6, 5+2) = (9,7), &nbsp;according to the rule of adding vectors.

Answer. &nbsp;The sum of the vectors &nbsp;(3,5)&nbsp; and &nbsp;(6,2)&nbsp; in the coordinate plane is the vector &nbsp;(9,7).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This &nbsp;Example&nbsp; is shown in &nbsp;Figure 1.

There is a special vector &nbsp;z = (0,0)&nbsp; in a coordinate plane. &nbsp;If you add this vector to any other vector &nbsp;u = (a,b), &nbsp;it will not change the vector &nbsp;u: &nbsp;u+z = (a,b) + (0,0) = (a,b) = u, &nbsp;according to the rule of adding vectors in a coordinate plane. &nbsp;So, &nbsp;regarding addition vectors in a coordinate plane, &nbsp;the vector &nbsp;z = (0,0)&nbsp; plays the role of the zero vector. &nbsp;This vector has zero projections in x- and y- axes, &nbsp;which means that its initial and terminal points coincide. &nbsp;This is consisted with the definition of the zero vector of the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>. &nbsp;The zero vector in a coordinate plane is denoted usually as &nbsp;0: &nbsp;0 = (0,0).

For any given vector &nbsp;u = (a,b)&nbsp; we can consider the vector &nbsp;v = (-a,-b). &nbsp;If you add these two vectors using the formula for adding vectors in a coordinate plane, &nbsp;you will get the zero vector: &nbsp;u+v = (a,b) + (-a,-b) = (a+(-a), b+(-b)) = (0,0) = 0. &nbsp;The vector &nbsp;v = (-a,-b)&nbsp; is called the &nbsp;opposite vector&nbsp; to the given vector &nbsp;u = (a,b)&nbsp; and is denoted as &nbsp;-u.

Now, recall these remarkable properties of adding vectors from the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>:

&nbsp;&nbsp;&nbsp;&nbsp;1) &nbsp;u + v = v + u &nbsp;for any two vectors &nbsp;u&nbsp; and &nbsp;v&nbsp; in a plane &nbsp;(commutativity property);

&nbsp;&nbsp;&nbsp;&nbsp;2) &nbsp;(u + v) + w = u + (v + w)&nbsp; for any three vectors &nbsp;u, &nbsp;v&nbsp; and &nbsp;w&nbsp; in a plane &nbsp;(associativity property);

&nbsp;&nbsp;&nbsp;&nbsp;3) &nbsp;u + 0 = u &nbsp;(existing and the key property of the zero vector);

&nbsp;&nbsp;&nbsp;&nbsp;4) &nbsp;u + (-u) = 0 &nbsp;(existing and the key property of the opposite vector).

Via the adding formula for vectors in a coordinate plane, &nbsp;these properties are hold for vectors in a coordinate plane too, &nbsp;and they are consistent with the similar properties of real numbers.

<H3>Subtracting vectors in a coordinate plane</H3>

Let &nbsp;PQ = u = (a,b) &nbsp;and &nbsp;PR = v = (c,d) &nbsp;be the two vectors in the coordinate plane &nbsp;(Figure 2).

P &nbsp; and &nbsp;R = (x3,y3) &nbsp;are the initial point and the terminal point of the vector &nbsp;v. &nbsp;Real numbers &nbsp;c = x3-x1&nbsp; and &nbsp;d = y3-y1&nbsp; are &nbsp;x- &nbsp;and &nbsp;y- projections of the vector &nbsp;PR.

We are going to calculate the difference &nbsp;u-v&nbsp; of these vectors.

The procedure of subtracting vectors in a plane is described in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>.

We suppose that the initial point of the vector &nbsp;u&nbsp; coincides with the initial point of the vector &nbsp;v &nbsp;as it is shown in &nbsp;Figure 2. &nbsp;If it is not so, &nbsp;we can make the parallel translation of the vector &nbsp;v&nbsp; to superpose its initial point onto the initial point of the vector &nbsp;u.

<TABLE>
 <TR>
 <TD>
{{{drawing( 282, 262, -2.0, 12.1, -2.0, 11.0,
 grid(1),
 line( 2.0, 2.0, 5.0, 7.0), 
 locate( 1.8, 1.9, P),
 locate( 3.3, 5.9, u),

line( 5.0, 7.0, 4.6,  6.6),
            line( 5.0, 7.0, 4.85, 6.5),

line( 2.0, 2.0, 11.0, 4.0),
          locate( 4.6, 7.8, Q),
          locate( 6.3, 3.0, v),

line( 11.0, 4.0,  10.5, 3.7),
            line( 11.0, 4.0,  10.5, 4.1),

red(line( 5.0, 7.0, 11.0, 4.0)),
            line( 5.0, 7.0, 5.6, 6.90),
            line( 5.0, 7.0, 5.4, 6.60),
          locate( 11.1, 4.4, R),
          locate( 8.5, 6.0, u-v),

locate( 1.8, 0.9, x1),
          locate( 4.8, 0.9, x2),
          locate(10.7, 0.9, x3),

locate( 0.3, 2.4, y1),
          locate( 0.3, 7.4, y2),
          locate( 0.3, 4.4, y3)
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 2. &nbsp;Subtracting vectors
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in a coordinate plane&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE>

According to the procedure of subtracting vectors of the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>, &nbsp;the difference of the vectors &nbsp;PQ&nbsp; and &nbsp;PR&nbsp; is the third side &nbsp;RQ&nbsp; of the triangle &nbsp;PQR&nbsp; with the initial point &nbsp;R&nbsp; and the terminal point &nbsp;Q.

The &nbsp;x-projection of the vector &nbsp;RQ&nbsp; is &nbsp;x2-x3. &nbsp;It is clear that &nbsp;x2-x3 = (x2-x1) - (x3-x1). &nbsp;Hence, &nbsp;the x-projection of the vector &nbsp;RQ&nbsp; is equal to &nbsp;a-c.

Similarly, the &nbsp;y-projection of the vector &nbsp;RQ&nbsp; is &nbsp;y2-y3. &nbsp;It is clear that &nbsp;y2-y3 = (y2-y1) - (y3-y1). &nbsp;Hence, &nbsp;the y-projection of the vector &nbsp;RQ&nbsp; is equal to &nbsp;b-d.

Thus the vector &nbsp;u-v&nbsp; has the &nbsp;x-projection equal to &nbsp;a-c &nbsp;and the &nbsp;y-projection equal to &nbsp;b-d. &nbsp;In the shortened component form, &nbsp;this means that &nbsp;(a,b) - (c,d) = (a-c,b-d).

This is the formula for subtracting vectors in a coordinate plane:<BLOCKQUOTE><TABLE BORDER=2>
 <TR>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;(a,b) - (c,d) = (a-c,b-d)&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE></BLOCKQUOTE>

Example 2. &nbsp;Find the difference of the vectors &nbsp;(3,5)&nbsp; and &nbsp;(9,2)&nbsp; in the coordinate plane.

Solution. &nbsp;We have &nbsp;(3,5) - (9,2) = (3-9, 5-2) = (-6,3), &nbsp;according to the rule of adding vectors.

Answer. &nbsp;The difference of the vectors &nbsp;(3,5)&nbsp; and &nbsp;(9,2)&nbsp; in the coordinate plane is the vector &nbsp;(-6,3).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This &nbsp;Example&nbsp; is shown in &nbsp;Figure 2.

<H3>Product of a vector by a real number in a coordinate plane</H3>

Let &nbsp;OP = u = (a,b) &nbsp;be the vector in the coordinate plane &nbsp;(Figure 3)&nbsp; and &nbsp;r&nbsp; be a real number.

As defined in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>, &nbsp;the product of the vector &nbsp;u&nbsp; by the real number &nbsp;r&nbsp; is the vector &nbsp;ru&nbsp; whose length is equal to &nbsp;|ru| = |r|*|u|&nbsp; and whose direction coincides with that of the vector &nbsp;u&nbsp; if r >= 0 &nbsp;or is opposite to the direction of the vector &nbsp;u&nbsp; if r < 0.

<TABLE>
 <TR>
 <TD>
{{{drawing( 282, 262, -2.0, 12.1, -2.0, 11.0,
 grid(1),
 line( 0.0, 0.0, 5.0, 4.0), 
 locate( 5.2, 4.1, P),
 locate( 2.3, 2.9, u),

line( 5.0, 4.0, 4.55, 3.80),
            line( 5.0, 4.0, 4.70, 3.55),

red(line( 5.0, 4.0, 10.0, 8.0)),
          locate( 10.2, 8.1, Q),
          locate( 7.1, 7.0, ru),

line( 10.0, 8.0,  9.55, 7.80),
            line( 10.0, 8.0,  9.70, 7.55),

green(line(0, 0, 5.0, 0)),
          locate( 2.3, 0.9, a),

blue(line(5.0, 0, 10.0, 0)),
          locate( 7.2, 0.9, ra),

green(line(0, 0,  0,   4.0)),
          locate(-1.4,  2.8, b),

blue(line(0, 4.0, 0, 8)),
          locate(-1.6, 6.8, rb)
)}}}

&nbsp;&nbsp;&nbsp;Figure 3. &nbsp;Product of the vector
by the real number in a coordinate plane&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE>

Let us consider the vector &nbsp;v = (ra,rb) &nbsp;and prove that it is equal to &nbsp;ru: &nbsp;(ra,rb) = ru.

The length of the vector &nbsp;u&nbsp; is equal to &nbsp;{{{sqrt(a^2+b^2)}}}. &nbsp;Hence, the length of the vector &nbsp;v&nbsp; is equal to &nbsp;{{{sqrt((ra)^2+(rb)^2)}}} = {{{r*sqrt(a^2+b^2)}}}, &nbsp;i.e.&nbsp; is &nbsp;r&nbsp; times the length of the vector &nbsp;u.

Next, if the real number &nbsp;r&nbsp; is positive then the vector &nbsp;v = (ra,rb)&nbsp; has the same direction as the vector &nbsp;u. &nbsp;Indeed, the ratio &nbsp;{{{rb/ra}}}&nbsp; is the same as the ratio &nbsp;{{{b/a}}}, &nbsp;hence, &nbsp;the vector &nbsp;v&nbsp; lies in the same straight line as the vector &nbsp;u. &nbsp;Since the components &nbsp;ra&nbsp; and &nbsp;rb&nbsp; have the same signs as the components &nbsp;a&nbsp; and &nbsp;b&nbsp; respectively, &nbsp;the vector &nbsp;v&nbsp; is co-directed with the vector &nbsp;u. &nbsp;Therefore, &nbsp;if the real number &nbsp;r&nbsp; is positive then the vector &nbsp;v = (ra,rb)&nbsp; satisfied the definition of the product the vector &nbsp;u&nbsp; by the number &nbsp;r.

If the real number &nbsp;r&nbsp; is negative then the vector &nbsp;v = (ra,rb)&nbsp; still lies in the same straight line as the vector &nbsp;u, but has the opposite direction as its components &nbsp;ra&nbsp; and &nbsp;rb&nbsp; have opposite signs to that of the vector &nbsp;u. &nbsp;In this case, &nbsp;the vector &nbsp;v = (ra,rb)&nbsp; satisfies again the definition of the product the vector &nbsp;u&nbsp; by the number &nbsp;r. Thus &nbsp;ru = (ra,rb).

This is the formula for product a vector by a real number in a coordinate plane:<BLOCKQUOTE><TABLE BORDER=2>
 <TR>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;r*(a,b) = (ra,rb)&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE></BLOCKQUOTE>

Example 3. &nbsp;Find the product of the vector &nbsp;(5,4)&nbsp; by the real number &nbsp;2&nbsp; in the coordinate plane.

Solution. &nbsp;We have &nbsp;2*(5,4) = (2*5, 2*4) = (10,8), &nbsp;according to the rule of product a vector by a number.

Answer. &nbsp;The product of the vector &nbsp;(5,4)&nbsp; by the real number 2 in the coordinate plane is the vector &nbsp;(10,8).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This &nbsp;Example&nbsp; is shown in &nbsp;Figure 3.

The operation of product of a vector by a real number in a coordinate plane has the following properties.

1) &nbsp;r(u + v) = ru + rv &nbsp;for any vectors &nbsp;u&nbsp; and &nbsp;v&nbsp; and for any real number &nbsp;r &nbsp;(distributivity of product with respect to addition of vectors).

&nbsp;&nbsp;&nbsp;&nbsp;Indeed, if &nbsp;u = (a,b)&nbsp; and &nbsp;v = (c,d)&nbsp; then &nbsp;r(u + v) = r*((a,b) + (c,d)) = r*(a+c,b+d) = (r*(a+c),r*(b+d)) = (ra+rc,rb+rd) = (ra,rb) + (rc,rd) = r(a,b) + r(c,d) = ru + rv.

2) &nbsp;(r+s)u = ru + su &nbsp;for any vector &nbsp;u&nbsp; and for any real numbers &nbsp;r&nbsp; and &nbsp;s &nbsp;(distributivity of product with respect to addition of constants).

&nbsp;&nbsp;&nbsp;&nbsp;Indeed, if &nbsp;u = (a,b)&nbsp; then &nbsp;(r+s)u = (r+s)*(a,b) = ((r+s)*a,(r+s)*b) = (r*a+s*a,r*b+s*b) = (r*a,r*b) + (s*a,s*b) = r*(a,b) + s*(c,d) = ru + su.

3) &nbsp;r(sa) = (rs)a &nbsp;for any vector &nbsp;a&nbsp; and for any real numbers &nbsp;r&nbsp; and &nbsp;s.

4) &nbsp;1a = a &nbsp;for any vector &nbsp;a &nbsp;(multiplication by unity).

<H3>Summary</H3>

1. &nbsp;If &nbsp;(a,b)&nbsp; and &nbsp;(c,d)&nbsp; are the vectors in the coordinate plane written in the shortened component form, &nbsp;then the sum of these vectors is &nbsp;(a+c,b+d).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This is the &nbsp;formula for adding vectors in a coordinate plane:<BLOCKQUOTE><TABLE BORDER=2>
 <TR>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;(a,b) + (c,d) = (a+c,b+d)&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE></BLOCKQUOTE>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Here &nbsp;a&nbsp; and &nbsp;b&nbsp; are &nbsp;x- and &nbsp;y- projections of the first vector &nbsp;(a,b), &nbsp;while &nbsp;c&nbsp; and &nbsp;d&nbsp; are &nbsp;x- and &nbsp;y- projections of the second vector &nbsp;(c,d).

2. &nbsp;If &nbsp;(a,b)&nbsp; and &nbsp;(c,d)&nbsp; are the vectors in the coordinate plane written in the shortened component form, &nbsp;then the difference of these vectors is &nbsp;(a-c,b-d).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This is the &nbsp;formula for subtracting vectors in a coordinate plane:<BLOCKQUOTE><TABLE BORDER=2>
 <TR>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;(a,b) - (c,d) = (a-c,b-d)&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE></BLOCKQUOTE>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Here &nbsp;a&nbsp; and &nbsp;b&nbsp; are &nbsp;x- and &nbsp;y- projections of the first vector &nbsp;(a,b), &nbsp;while &nbsp;c&nbsp; and &nbsp;d&nbsp; are &nbsp;x- and &nbsp;y- projections of the second vector &nbsp;(c,d).

3. &nbsp;If &nbsp;(a,b)&nbsp; is the vector in the coordinate plane written in the shortened component form, &nbsp;then the product of this vector by a real number &nbsp;r&nbsp; is &nbsp;(ra,rb).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This is the &nbsp;formula for product a vector by a number in a coordinate plane:<BLOCKQUOTE><TABLE BORDER=2>
 <TR>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;r*(a,b) = (ra,rb)&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE></BLOCKQUOTE>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Here &nbsp;a&nbsp; and &nbsp;b&nbsp; are &nbsp;x- and &nbsp;y- projections of the vector &nbsp;(a,b).

4. &nbsp;The set of vectors as geometric objects defined in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>&nbsp; and the set of vectors in coordinate plane written in the component form as defined in the

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-coordinate-plane.lesson>Vectors in a coordinate plane</A>&nbsp; are in one-to-one correspondence: &nbsp;the vector as the geometric object corresponds to the pair of its projections to &nbsp;x- &nbsp;and &nbsp;y- axes.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This correspondence is consisted with the operations adding and subtracting vectors, as well as multiplying vectors by numbers.

My introductory lessons on vectors in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-plane.lesson>Vectors in a plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-that-are-coherently-oriented-sides-of-a-convex-closed-polygon.lesson>Sum of vectors that are coherently oriented sides of a convex closed polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-that-are-coherently-oriented-sides-of-an-unclosed-polygon.lesson>Sum of vectors that are coherently oriented sides of an unclosed polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-that-connect-the-center-of-a-parallelogram-with-its-vertices.lesson>Sum of vectors that connect the center of a parallelogram with its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Vectors-in-a-coordinate-plane.lesson>Vectors in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- Addition, Subtraction and Multiplication by a number of vectors in a coordinate plane&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(this lesson)

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Summing-vectors-that-are-coherently-oriented-sides-of-a-convex-closed-polygon.lesson>Summing vectors that are coherently oriented sides of a convex closed polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Summing-vectors-that-are-coherently-oriented-sides-of-an-unclosed-polygon.lesson>Summing vectors that are coherently oriented sides of an unclosed polygon</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-Centroid-of-a-triangle-is-the-Intersection-point-of-its-medians.lesson>The Centroid of a triangle is the Intersection point of its medians</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-Centroid-of-a-parallelogram-is-the-Intersection-point-of-its-diagonals.lesson>The Centroid of a parallelogram is the Intersection point of its diagonals</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Sum-of-the-vectors-connecting-the-center-of-mass-of-a-triangle-with-its-vertices.lesson>Sum of vectors connecting the center of mass of a triangle with its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Sum-of-the-vectors-connecting-the-center-of-mass-of-a-quadrilateral-with-its-vertices.lesson>Sum of vectors connecting the center of mass of a quadrilateral with its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-the-vectors-from-the-center-of-mass-of-the-n-sided-polygon-to-its-vertices.lesson>Sum of vectors connecting the center of mass of a n-sided polygon with its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Sum-of-vectors-connecting-the-center-of-a-regular-polygon-with-its-vertices.lesson>Sum of vectors connecting the center of a regular n-sided polygon with its vertices</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Solved-problems-on-vectors-in-a-plane.lesson>Solved problems on vectors in a plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Vectors/Solved-problems-on-vectors-in-a-coordinate-plane.lesson>Solved problems on vectors in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-find-the-length-of-the-vector-in-a-coordinate-plane.lesson>HOW TO find the length of the vector in a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Vectors/Flying-airplane-blowing-wind-airspeed-groundspeed-etc.lesson>Flying airplane, blowing wind, airspeed, groundspeed etc.</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Vectors/REVIEW-of-Introductory-lessons-on-vectors-in-a-plane.lesson>OVERVIEW of Introductory lessons on vectors in a plane</A>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.