7.5 Scalar and Vector Projections
Chapter
Chapter 7
Section
7.5
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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?

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0.49mins
Q1a

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

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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}.

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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}?

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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}.

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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}.

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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)

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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)

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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)

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1.04mins
Q7c

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

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1.07mins
Q8a

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

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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.

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0.49mins
Q9

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

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0.42mins
Q10a

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

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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).

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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)?

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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}.

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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}?

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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).

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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}.

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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}.

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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.

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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

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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.

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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}?

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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

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1.44mins
Q16

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

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1.12mins
Q17