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

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Addition, subtraction and multiplication by a number of vectors in a coordinate plane

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

The lesson is the continuation of the lessons Vectors in a plane and Vectors in a coordinate plane under the current topic in this site.

Recall that in a coordinate plane each vector u can be presented in the component form u = (a,b), where the real numbers a and b are the projections of the vector u
to the coordinate axes x and y respectively (see the lesson Vectors in a coordinate plane).

Adding vectors in a coordinate plane

Let PQ = u = (a,b) and QR = v = (c,d) be the two vectors in the coordinate plane (Figure 1).
P = (x1,y1) and Q = (x2,y2) are the initial point and the terminal point of the vector u. Real numbers a = x2-x1 and b = y2-y1 are x- and y- projections of the vector PQ.
Q and R = (x3,y3) are the initial point and the terminal point of the vector v. Real numbers c = x3-x2 and d = y3-y2 are x- and y- projections of the vector QR.
We are going to calculate the sum u+v of these vectors.
The procedure of adding vectors in a plane is described in the lesson Vectors in a plane (the triangle rule or the parallelogram rule).

We suppose that the terminal point of the vector u coincides with the initial point of the vector v as it is shown in Figure 1. If it is not so, we can make the parallel translation of the vector v
to superpose its initial point onto the terminal point of the vector u, according to the procedure adding vectors of the lesson Vectors in a plane.

Figure 1. Adding vectors
in a coordinate plane (1)

According to the triangle rule of adding vectors, the sum of the vectors PQ and QR is the third side PR of the triangle PQR with the initial point P and the terminal point R. The x-projection of the vector PR is x3-x1. It is clear that x3-x1 = (x3-x2) + (x2-x1). In other words, the x-projection of the vector QR is equal to a+c.
Similarly, the y-projection of the vector PR is y3-y1. Again, it is clear that y3-y1 = (y3-y2) + (y2-y1). In other words, the y-projection of the vector QR is equal to b+d.
Thus the vector u+v has the x-projection equal to a+c and the y-projection equal to b+d. In the shortened component form, this means that (a,b) + (c,d) = (a+c,b+d).

This is the formula for adding vectors in a coordinate plane:

(a,b) + (c,d) = (a+c,b+d)

One can derive this rule geometrically, too, by playing with the projection segments in x- and y- axes that are shown in the Figure 1 in green and blue colors. The geometric proof is based on the fact that the co-named projection segments of the vectors u and v join each other. Such a proof should consider different dispositions of the vectors u and v that create positive and negatives values of the projections. The algebraic proof provided above takes into account all these details automatically.

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

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

Answer. The sum of the vectors (3,5) and (6,2) in the coordinate plane is the vector (9,7).
                This Example is shown in Figure 1.

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

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

Now, recall these remarkable properties of adding vectors from the lesson Vectors in a plane:

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

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

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

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

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

Subtracting vectors in a coordinate plane

Let PQ = u = (a,b) and PR = v = (c,d) be the two vectors in the coordinate plane (Figure 2).
P = (x1,y1) and Q = (x2,y2) are the initial point and the terminal point of the vector u. Real numbers a = x2-x1 and b = y2-y1 are x- and y- projections of the vector PQ.
P and R = (x3,y3) are the initial point and the terminal point of the vector v. Real numbers c = x3-x1 and d = y3-y1 are x- and y- projections of the vector PR.
We are going to calculate the difference u-v of these vectors.
The procedure of subtracting vectors in a plane is described in the lesson Vectors in a plane.

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

Figure 2. Subtracting vectors
in a coordinate plane

According to the procedure of subtracting vectors of the lesson Vectors in a plane, the difference of the vectors PQ and PR is the third side RQ of the triangle PQR with the initial point R and the terminal point Q.
The x-projection of the vector RQ is x2-x3. It is clear that x2-x3 = (x2-x1) - (x3-x1). Hence, the x-projection of the vector RQ is equal to a-c.
Similarly, the y-projection of the vector RQ is y2-y3. It is clear that y2-y3 = (y2-y1) - (y3-y1). Hence, the y-projection of the vector RQ is equal to b-d.
Thus the vector u-v has the x-projection equal to a-c and the y-projection equal to b-d. In the shortened component form, this means that (a,b) - (c,d) = (a-c,b-d).

This is the formula for subtracting vectors in a coordinate plane:

(a,b) - (c,d) = (a-c,b-d)

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

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

Answer. The difference of the vectors (3,5) and (9,2) in the coordinate plane is the vector (-6,3).
This Example is shown in Figure 2.

Product of a vector by a real number in a coordinate plane

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

As defined in the lesson Vectors in a plane, the product of the vector u by the real number r is the vector ru whose length is equal to |ru| = |r|*|u| and whose direction coincides with that of the vector u if r >= 0 or is opposite to the direction of the vector u if r < 0.

Figure 3. Product of the vector
by the real number in a coordinate plane

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

The length of the vector u is equal to sqrt%28a%5E2%2Bb%5E2%29

. Hence, the length of the vector v is equal to sqrt%28%28ra%29%5E2%2B%28rb%29%5E2%29

, i.e. is r times the length of the vector u.

Next, if the real number r is positive then the vector v = (ra,rb) has the same direction as the vector u. Indeed, the ratio rb%2Fra

is the same as the ratio b%2Fa

, hence, the vector v lies in the same straight line as the vector u. Since the components ra and rb have the same signs as the components a and b respectively, the vector v is co-directed with the vector u. Therefore, if the real number r is positive then the vector v = (ra,rb) satisfied the definition of the product the vector u by the number r.
If the real number r is negative then the vector v = (ra,rb) still lies in the same straight line as the vector u, but has the opposite direction as its components ra and rb have opposite signs to that of the vector u. In this case, the vector v = (ra,rb) satisfies again the definition of the product the vector u by the number r. Thus ru = (ra,rb).

This is the formula for product a vector by a real number in a coordinate plane:

r*(a,b) = (ra,rb)

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

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

Answer. The product of the vector (5,4) by the real number 2 in the coordinate plane is the vector (10,8).
                This Example is shown in Figure 3.

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

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

    Indeed, if u = (a,b) and v = (c,d) then 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) (r+s)u = ru + su for any vector u and for any real numbers r and s (distributivity of product with respect to addition of constants).

    Indeed, if u = (a,b) then (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) r(sa) = (rs)a for any vector a and for any real numbers r and s.

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

Summary

1. If (a,b) and (c,d) are the vectors in the coordinate plane written in the shortened component form, then the sum of these vectors is (a+c,b+d).
This is the formula for adding vectors in a coordinate plane:

(a,b) + (c,d) = (a+c,b+d)

Here a and b are x- and y- projections of the first vector (a,b), while c and d are x- and y- projections of the second vector (c,d).

2. If (a,b) and (c,d) are the vectors in the coordinate plane written in the shortened component form, then the difference of these vectors is (a-c,b-d).
This is the formula for subtracting vectors in a coordinate plane:

(a,b) - (c,d) = (a-c,b-d)

Here a and b are x- and y- projections of the first vector (a,b), while c and d are x- and y- projections of the second vector (c,d).

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

r*(a,b) = (ra,rb)

     Here a and b are x- and y- projections of the vector (a,b).

4. The set of vectors as geometric objects defined in the lesson Vectors in a plane and the set of vectors in coordinate plane written in the component form as defined in the
     lesson Vectors in a coordinate plane are in one-to-one correspondence: the vector as the geometric object corresponds to the pair of its projections to x- and y- axes.

     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
    - Vectors in a plane
    - Sum of vectors that are coherently oriented sides of a convex closed polygon
    - Sum of vectors that are coherently oriented sides of an unclosed polygon
    - Sum of vectors that connect the center of a parallelogram with its vertices
    - Vectors in a coordinate plane
    - Addition, Subtraction and Multiplication by a number of vectors in a coordinate plane               (this lesson)
    - Summing vectors that are coherently oriented sides of a convex closed polygon
    - Summing vectors that are coherently oriented sides of an unclosed polygon
    - The Centroid of a triangle is the Intersection point of its medians
    - The Centroid of a parallelogram is the Intersection point of its diagonals
    - Sum of vectors connecting the center of mass of a triangle with its vertices
    - Sum of vectors connecting the center of mass of a quadrilateral with its vertices
    - Sum of vectors connecting the center of mass of a n-sided polygon with its vertices
    - Sum of vectors connecting the center of a regular n-sided polygon with its vertices
    - Solved problems on vectors in a plane
    - Solved problems on vectors in a coordinate plane
    - HOW TO find the length of the vector in a coordinate plane
    - Flying airplane, blowing wind, airspeed, groundspeed etc.

    - OVERVIEW of Introductory lessons on vectors in a plane

Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II.

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