The vector \vec{a} = (2, 3) is projected onto the x-axis. What is the scalar projection? What is the vector projection?
What are the scalar and vector projections when \vec{a} = (2, 3) is projected onto the y-axis?
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}.
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}?
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}.
(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}.
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)
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)
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)
Determine the scalar and vector projections of \vec{a} =(-1, 2, 4) on each of the three axes.
What are the scalar and vector projections of m(-1, 2, 4) on each of the three axes?
(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.
Using a diagram, show that the vector projection of -\vec{a} on \vec{a} is -\vec{a}.
Using the formula for determining scalar projections, show that the result in part a. is true.
(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).
(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)?
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}.
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}?
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).
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}.
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}.
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.
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
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.
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}?
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