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VC.01 Vectors Point The Way

Literacy
1. When you plot a certain 3D vector X with its tail at {0, 0, 0}, it turns out that the tip of X is sitting
on {9, -6, 4}. When you plot the same vector with its tail at {-3, 5, 7}, where will the tip of X be?

2. Given two 2D vectors X and Y, what do you get when you plot Y and X+Y with their tails at the
origin, and you plot X with its tail at the tip of Y?

Illustrate with a sketch.

What happens in 3D?

3. Given two 2D vectors X and Y, what do you get when you plot X and X-Y with their tails at the
origin, and you plot Y with its tail at the tip of X-Y?

Illustrate with a sketch.

What happens in 3D?

4.
a) Given X = {-1, 2, 5}, calculate 3X.

b) Can you change the direction of a vector by multiplying it by a positive number?

c) What happens when you multiply a vector by a negative number?

d) Give a unit vector in the OPPOSITE direction of X.

5.
a) Calculate X.Y for X = {2, 7, 1} and Y = {2, -2, 5}

b) What does this result tell you about these two vectors?

6.
a) Calculate X.Y for X = {-1, 4, 5} and Y = {2, -2, 2}

b) What does this result tell you about these two vectors?

7. Give the number t that makes X = {3, -4, 2} and Y = {2, -3, t} orthogonal
8. Here are parametric formulas for two 3D lines:
L1[t ] = {3, 0,1} + t {−2,1,1}
L2 [t ] = {3, 0,1} + t {1,1,1}
Say how you can tell that these lines cross each other at right angles.

9. You are walking along a curve and at time t you are at the location P[t] = {x[t], y[t], z[t]}. You stop
at a certain time t 0 and plunk down the velocity vector P ′[t 0 ] = {x ′[t 0 ], y ′[t 0 ], z ′[t 0 ]} so that its tail is
right on the point P [t 0 ] . Does the resulting tangent vector point forward in the direction you are
going, or does it point back against the direction you are walking?

 X .Y 
10. The component of a vector X in the direction of another vector is given by  Y .
 Y .Y 
What does this formula reduce to if Y is a unit vector?

1 1   t t 
11a. Put the tail of the vector t  ,  =  ,  at {0, 0} and say what t must be to make the tip of this
2 2 2 2
vector as close to {1, 3} as it can be.

11b. Write B as the sum of a vector parallel to A and a vector othorgonal to A. B = 3j + 4k, A = i + j
  1 1 
12. Here are the vectors X =  ,  and Y = {1, 3.5} with their tailsyat {0, 0}.
 2 2 4

a) Identify X and Y in the picture

b) Pencil in the component of Y in the direction of X 3

c) Calculate the component of Y in the direction of X

0 x
0.0 0.5 1.0 1.5 2.0

 1
13. Here are the vectors X = 1,  and Y = {-3, 1} with their tails at {0, 0}.
 2
y
2
a) Identify X and Y in the picture

b) Pencil in the component of Y in the direction of X

1

c) Calculate the component of Y in the direction of X

x
3 2 1 1 2

1

2

{ }
14. At time t with t ≥ 0 , an object is at the position P[t ] = Sin [2t ], ln [3t ], e −3 t .
Calculate its velocity, vel[t], acceleration, accel[t], and speed, speed[t], as functions of t
15. Here is a plot of an object’s path shown with two acceleration vectors with their tails at the points
on the curve at which they are calculated: y
3

The object is moving of this curve from lower left to upper right.

a) Pencil in the tangential and normal components of each of

2
the two acceleration vectors in the plot.

b) At which of these points is the speed increasing?

1

c) At which of these points is the speed decreasing? x

2.5 3.0 3.5 4.0 4.5

1

16. Ballistic projectiles (like cannonballs from a cannon) fired from the origin with muzzle velocity
v0 ft / sec and angle b with the horizontal are at the position
{ }
P [t ] = v0 Cos[b] t , v0 Sin[b] t − 16t 2 , t seconds after firing

a) Calculate the velocity vector as a function of t.

b) Calculate the acceleration vector as a function of t.

c) Explain the result of the acceleration vector.

17.
a) If X.Y > 0, then is the push of X in the direction of Y with Y or against Y?

b) If X.Y < 0, then is the push of X in the direction of Y with Y or against Y?

18. Anywhere you happen to be, you feel a push whose direction is the same as the direction of the
{ }
vector {2, 1}. Your velocity at time t > 0 is given by vel[t ] = − t , t 2 .

a) For what times t > 0 are you being helped by the push?

b) At what times t > 0 are you being hindered by the push?

19.
a) Give parametric equations for the line that passes through the points {3, 1} and {5, 13}

b) Give a vector parallel to this line.

c) Give a vector perpendicular to this line.

20. Are the lines with parametric formulas

L1[t ] = {2, 3} + t {−2, 4}
L2 [t ] = {2, 3} + t {−4, 8}
the same or different lines? You must give a full, convincing argument.

21. Are the lines with parametric formulas

L1[t ] = {2, 3} + t {−3, 5}
L2 [t ] = {−4,13} + t {−3, 5}
the same or different lines? You must give a full, convincing argument.
22. Give parametric equations for the 3D line through the points {2, 5, 4} and {4, -1, 2}

2
1
23. Here are two parallel circles: 0

1
a) When you fix a given s with 0 ≤ s ≤ 2π ,
2
then what happens to the points
{2Cos[ s], 2Sin[s ], 0} + t ({Cos[s ], Sin[s ], 3} − {2Cos[ s], 2Sin[s ], 0})
3

as you run t from 0 to 1? 2

Explain and draw the result in the picture to the right. 1

0
2
1
0
1
2

b) Use your answer to predict the output from the following mathematica code:

ParametricPlot3D[{2 Cos[s],2 Sin[s],0}+ t ({Cos[s],Sin[s],3}-{2 Cos[s],2 Sin[s],0}),{s,0,2 π},{t,0,1}]

Explain and draw a sketch of the output.

24. Derive the equation X .Y = X Y Cosϑ