7.5 Scalar and Vector Projections
Chapter
Chapter 7
Section
7.5
Lectures 6 Videos
Solutions 27 Videos

The vector \vec{a} = (2, 3) is projected onto the x-axis. What is the scalar projection? What is the vector projection?

0.49mins
Q1a

What are the scalar and vector projections when \vec{a} = (2, 3) is projected onto the y-axis?

0.26mins
Q1b

Explain why it is not possible to obtain either a scalar projection or a vector projection when a nonzero vector \vec{x} is projected on \vec{0}.

0.23mins
Q2

Consider two nonzero vectors, \vec{a} and \vec{b}, that are perpendicular to each other. Explain why the scalar and vector projections of \vec{a} on \vec{b} must be 0 and \vec{0}, respectively. What are the scalar and vector projections of \vec{b} on \vec{a}?

0.31mins
Q3

Draw two vectors, \vec{p} and \vec{q}. Draw the scalar and vector projections of \vec{p} on \vec{q}.

Show, using your diagram, that these projections are not necessarily the same as the scalar and vector projections of \vec{q} on \vec{p}.

2.20mins
Q5

(a) For the vectors \vec{p} = (3, 6, -22) and \vec{q} = (-4, 5, -20), determine the scalar and vector projections of \vec{p} on \vec{q}.

(b) Determine the direction angles for \vec{p}.

2.01mins
Q6

For each of the following, determine the scalar and vector projections of \vec{x} on \vec{y}.

\vec{x} = (1, 1), \vec{y} = (1, -1)

0.45mins
Q7a

For each of the following, determine the scalar and vector projections of \vec{x} on \vec{y}.

\vec{x} = (2, 2\sqrt{x}), \vec{y} = (1, 0)

0.37mins
Q7b

For each of the following, determine the scalar and vector projections of \vec{x} on \vec{y}.

\vec{x} = (2, 5), \vec{y} = (-5, 12)

1.04mins
Q7c

Determine the scalar and vector projections of \vec{a} =(-1, 2, 4) on each of the three axes.

1.07mins
Q8a

What are the scalar and vector projections of m(-1, 2, 4) on each of the three axes?

0.54mins
Q8b

(a) Given the vector $\vec{a}$, show with a diagram that the vector projection of $\vec{a}$ on$\vec{a}$is$\vec{a}$and that the scalar projection of$\vec{a}$on$\vec{a}$is$|\vec{a}|$

(b) Using the formulas for scalar and vector projections, explain why the results in part a. are correct if we use u 0^{\circ} for the angle between the two vectors.

0.49mins
Q9

Using a diagram, show that the vector projection of -\vec{a} on \vec{a} is -\vec{a}.

0.42mins
Q10a

Using the formula for determining scalar projections, show that the result in part a. is true.

0.55mins
Q10b

(a) Find the scalar vector projections of \vec{AB} along each of the axes if A has coordinates (1, 2, 2) and B has coordinates (-1, 3, 4).

1.22mins
Q11a

(b) What angle does \vec{AB} make with the y-axis if A has coordinates (1, 2, 2) and B has coordinates (-1, 3, 4)?

0.38mins
Q11b

In the diagram shown, \triangle ABC is an isosceles triangle where |\vec{a}| = |\vec{b}|.

a) Draw the scalar projection of \vec{a} on \vec{c}.

0.26mins
Q12a

In the diagram shown, \triangle ABC is an isosceles triangle where |\vec{a}| = |\vec{b}|.

(b) Relocate \vec{b}, and draw the scalar projection of \vec{b} on \vec{c}.

(c) Explain why the scalar projection of \vec{a} on \vec{c} is the same as the scalar projection of \vec{b} on \vec{c}.

(d) Does the vector projection of \vec{a} on \vec{c} equal the vector projection of \vec{b} on \vec{c}?

0.48mins
Q12bcd

Vectors \vec{a} and \vec{b} are such that |\vec{a}|= 10 and |\vec{b}| = 12, and the angle between them is 135^{\circ}.

a) Show that the scalar projection of \vec{a} on \vec{b} does not equal the scalar projection of \vec{b} on \vec{a}.

b) Draw diagrams to illustrate the corresponding vector projections associated with part a).

0.45mins
Q13

You are given the vector \vec{OD} = (-1, 2, 2) and the three points, A(-2, 1, 4), B(1, 3, 3), and C(-6, 7, 5).

(a) Calculate the scalar projection of \vec{AB} on \vec{OD}.

1.25mins
Q14a

You are given the vector \vec{OD} = (-1, 2, 2) and the three points, A(-2, 1, 4), B(1, 3, 3), and C(-6, 7, 5).

Verify computationally that the scalar projection \vec{BC} on \vec{OD} equal the scalar projection of \vec{AC} on \vec{OD}.

3.20mins
Q14b

You are given the vector \vec{OD} = (-1, 2, 2) and the three points, A(-2, 1, 4), B(1, 3, 3), and C(-6, 7, 5).

Explain why this same result is also true for the corresponding vector projections.

1.56mins
Q14c

If \alpha, \beta and \gamma represent the direction angles for vector \vec{OP}, prove that \cos^2\alpha + \cos^2 \beta + \cos^2 \gamma = 1

2.27mins
Q15a

Determine the coordinates of a vector \vec{OP} that makes an angle of 30^{\circ} with y-axis, 60^{\circ} with the z-axis, and 90^{\circ} with the x-axis.

1.34mins
Q15b

In Example 3, it was shown that, in general, the direction angles do not always add to 180^{\circ} -that is, \alpha+ \beta + \gamma \neq 180^{\circ}. Under what conditions, however, must the direction angles always add to 180^{\circ}?

1.21mins
Q15c

A vector in \mathbb{R}^3 makes equal angles with the coordinate axes. Determine the size of each of these angles if the angles are

a. acute

b. obtuse

If \alpha, \beta and \gamma represent the direction angles for vector \vec{OP}, prove that \cos^2\alpha + \cos^2 \beta + \cos^2 \gamma = 2`