Questions on Algebra: Introduction to vectors, addition and scaling answered by real tutors!

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Question 1201561: Pascal and his friend Fermat live on opposite sides of a river that is 1 km wide. Fermat lives 2 km downstream from Pascal on the opposite side of the river. Pascal can swim at a rate of 3 km/h and the river’s current has a speed of 4 km/h. Pascal swims from his cottage directly across the river.
a) What is Pascal’s resultant velocity?
b) How far away from Fermat’s cottage will Pascal be when he reaches the other side?
c) How long will it take Pascal to reach the other side?
please help me with this vectors question. it's from my "vectors as veolcity" lesson. thank you so much.
Click here to see answer by ikleyn(52778)

Question 1201557: hello. this is from my 'spanning sets' lesson from vectors:
the set of vectors {(1,0,0), (0,1,0)} spans a set in R3
a. describe the set
b. write the vector (-2, 4, 0) as a linear combination of these vectors
c. explain why it is not possible to write ( 3,5,8) as a linear combination of these vectors
d. If we added the vector (1,1,0) to this set, would it now span R3? Explain.
thank you.
Click here to see answer by ikleyn(52778)

Question 1201656: If (-1,3,5) x (0,a,1) = (-2,1,-1), determine a
This is from my 'cross product' lesson under vectors. Thank you.
Click here to see answer by ikleyn(52778)

Question 1201856: 'Lines in 3-Space' lesson:
1. State where possible vector, parametric, and symmetric equations for each of the following lines.
a. The line passing through the point P (-1,2,1) with direction vector (3,-2,1)
b. The line passing through the point B (-2,3,0) and parallel to the line passing through the points M (-2,-2,1) and N (-2,4,7)
c.The line passing through the points Q (1,2,4) and parallel to the z-axis
Click here to see answer by ikleyn(52778)

Question 1201855: from my 'Vector Equations of Planes' lesson
1.
the plane with the equation (vector) r = (1,2,3)+ s (1,2,5) + t(1,-1,3) where s, t E R inrersects the y and z-axes at the points A and B. Determine the equation of the line thru the points A and B.
Click here to see answer by math_tutor2020(3816)

Question 1201854: from "Vector Equations of Planes" lesson.
A plane passes through the points P(-2,3,1), Q(-2,3,2) and R (1,0,1)
a. using vectors PQ and vectors PR as your direction vectors, write the vector equation of this plane
b. using vector QR, and any other direction vector, write a second vector equation for this plane.
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Question 1201853: this is from my 'Cartesian Equation of a Plane' lesson.
a. Determine the Cartesian equation of the plane that contains the three points A(-2,3,1), B(3,4,5) and C(1,1,0).

b. Determine the Cartesian equation of the plane containing the point P(-1,1,0) and perpendicular to the line between the points A(1,2,1) and B(3,-2,0).
thank you.
Click here to see answer by Alan3354(69443)

Question 1201857: from my 'Vector Equation of a Line' lesson
A line has the equation y = (-5/6)x + 9
a. Give a direction vector for a line that is parallel to this line.
b. Give a direction vector for a line that is perpendicular to this line.
c. Give the coordinates of a point on the given line.
d. In both vector and parametric form, give the equations of the line parallel to the given line and passing through A (7,9)
e. In both vector and parametric form, give the equations of the line perpendicular to the given line and passing through B (-2,1)
Click here to see answer by math_tutor2020(3816)

Question 1202170: Determine the vector equation of the line that passes through the point A (-2, 3, 6) and is parallel to the line of intersection between the two planes
pi 1: 2x - y + z = 0 and
pi 2: y + 4z = 0
the answer is supposed to be vector r2 = (-2, 3, 6) + s (-5, -8, 2) sER
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Question 1202475: What unit vector is in the same direction as v = 2i - 4j?
Show all work.
Click here to see answer by ikleyn(52778)

Question 1202476: What unit vector is in the same direction as v = 2i - 4j?
Show all work.

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Question 1202643: given that a=2i+3j-k, b=i-j+2k, and c=3i+4j+k, find (a) a+2b-c, (b) a vector d such that a+b+c+d=0, and (c) a vector d such that a-b+c+3d=0
Click here to see answer by Alan3354(69443)

Question 1203102: Determine whether the lines L1 and L2 are parallel, skew, or intersecting.

L1: x = 6+4t, y = 8-2t, z = 2+6t
L2: x = 4+16s, y = 12-8s, z = 16+20s

Click here to see answer by ikleyn(52778)

Question 1203564: given that Avector=3i+4j+8k,Bvector=2i+4j-6k,and Cvector=i-3j+4k
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Question 1203564: given that Avector=3i+4j+8k,Bvector=2i+4j-6k,and Cvector=i-3j+4k
Click here to see answer by ikleyn(52778)

Question 1203668: Let e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) ∈ R 3. Find all real numbers c ∈ R such that the angle between the vectors −e1 + 2e2 + ke3 and −e1 + ke2 + 2e3 is π/2 (they are orthogonal).

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Question 1203666: Show that 4u · v = ||u + v||^2 − ||u − v||^2. Show that u and v is orthogonal if and only if ||u + v|| = ||u − v||.
Click here to see answer by ikleyn(52778)

Question 1203665: Show that 4u · v = ||u + v||^2 − ||u − v||^2. Show that u and v is orthogonal if and
only if ||u + v|| = ||u − v||.

Click here to see answer by MathLover1(20849)

Question 1203667: Let u, v ∈ R^n be vectors in R^n
(a) State and prove the Cauchy - Schwarz inequality. Give a necessary and sufficient
condition on u, v ∈ R^n such that the equality holds.
(b) State and prove the triangle inequality. Show that the equality holds if and only if u is a scalar multiple of v.
Click here to see answer by ikleyn(52778)

Question 1203884: Dora was travelling a = 6km, 30° N of E, b = 3km, 60° S of E, and then c = 10km, 15° N of E. At the end of this trek, she realized that in the last leg she should have travelled c' = 4km, 75° W of N instead of the third path she originally took. In component form, what path should Dora take to reach her supposed destination? What is the displacement from her initial position?
Click here to see answer by Theo(13342)

Question 1204008: a hiker walks 14.7 km at an angle of 305 from east. find the x and y component of this walk. Include the diagram.
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Question 1204113: An airplane is flying at 340 km/hr at 12 degrees of East of North. The wind is blowing 40 km/hr at 34 degrees South of East. What is the planes actual velocity?
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Question 1205871: Considering AC and BC to be carrying 16kn and 12kn respectively, what is the magnitude and the direction of the resultant? What is the magnitude and the direction of a single vector, that can zero out the resultant?
Click here to see answer by ikleyn(52778)

Question 1205872: You are standing on the ground at the origin of a coordinate system.An airplane flies over you with a constant velocity parallel to the x-axis with constant velocity parallel to the x-axis at a fixed height of 8.30×10³m. At a reference time, t=0 seconds, the airplane may be assumed to be directly above you such that the position vector is P0=(0.00×10³m)î + 8.30×10³m) j. Determine the magnitude and orientation of the airplanes position vector another 30 seconds, P50.
Click here to see answer by ikleyn(52778)

Question 1206226: Three vectors add together so that the resultant is zero. Vector A points 75.0° north of East. Vector B points due west. Vector C points due south and has magnitude of 185 metres. Find the magnitude of A and B.
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Question 1206230: A vector has magnitude 6.0 units due east, vector B points due north. Find
a) the magnitude of B if A+B points 60° north of east?
b) the magnitude of A+B
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Question 1206237: Show that (AxB).(AxB) + (A.B)^2 =(A^2)(B^2)
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Question 1206236: Prove that two vectors must have equal magnitudes if their sum is perpendicular to their difference
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Question 1206236: Prove that two vectors must have equal magnitudes if their sum is perpendicular to their difference
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Question 1206238: A ladder AB rests against a vertical wall 0A. The foot 'B' of the ladder is pulled away with constant speed V
a) show that the midpoint of the ladder describes the arc of a circle of radius a/2
b) Find the velocity and speed of the midpoint of the ladder at the instant B is distant b
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Question 1206234: Two vectors a and B have components in arbitrary units ax=3.2, ay= 1.6, bx=0.50, by=4.5
a) find the angle between a and B
b) Find the components of a vector C that is perpendicular to a and is in the x-y plane, and has magnitude of 5.0 units
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Question 1206235: Vector A lies in the y-z plane 63° from the positive y-axis and has a magnitude 3.2. Vector B lies in the x-z plane 48° from the x-axis and has a magnitude 1.4.
Find A.B, AxB and the angle between A and B
Click here to see answer by Edwin McCravy(20054)

Question 1207272: [p] = 3 [q] = 4
p . q = 10
Find the value of
q.(p + q)
I am unsure of the square bracket around the letters p and q.
Multiplied out:-
q.p + q^2
4 x 3 + 4^2
= 28
but answer is 26!
Can you help me please?
Click here to see answer by math_tutor2020(3816)

Question 1199547: A point travels as described by the following parametric equations x=10t+10cos3.14t, y=20t+10sin3.14t and z=30t, where x,y,z, are in meters, t in seconds, all angles in radians. The vector locating the body at any time is r=ix+jy+kz. Determine the magnitude of the velocity of the body in meters per second at time t = 0.75 second.
Click here to see answer by CPhill(1959)

Question 1185368: A hockey puck of mass 0.245 kg leaves a hockey stick at a speed of 29 m/s. The coefficient of kinetic friction between the ice and puck is 0.080.
What is the acceleration of the puck once it leaves the stick?
How long will it take for the puck to stop?
Repeat part (a) for a puck of twice the mass.

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Question 1179249: Find value of the folllwing:
(a) (a. b) + (a x b)²
(b) (a. b) + (a x 2b)²
.
Note: Here . means dot product and x means cross or vector product.
Click here to see answer by CPhill(1959)

Question 1178892: What is the correct way to write the following function:
K(x, xi) = Square of alpha multiplied with exp power minus of summation of k of (x - xi)² if we have following condition:
..
Vectors x1, x2, x3, ... , xn belongs to d dimensional Real space (R with superscript d) and d is from 1 to D dimensional.
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Question 1170575: Find the vector equation for the line of intersection of the planes 4x-y+2z=0 and 4x+z=3
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Question 1167891: A swimmer is capable of swimming at 1.4m/s in still water
(A) how far down the stream will he land y he swims directly across a 180m wide river
(B) how long will it take him to reach the other side
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Question 1165416: Find vector equation for the line with the following cartesian equation
(a)
X+3y=7
(b)
2x-5y=3
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