Chapter 3

A protester carries his sign of protest, starting from the origin of an \(x y z\) coordinate system, with the \(x y\) plane horizontal. He moves \(40 \mathrm{~m}\) in the negative direction of the \(x\) axis, then \(20 \mathrm{~m}\) along a perpendicular path to his left, and then \(25 \mathrm{~m}\) up a water tower. (a) In unit-vector notation, what is the displacement of the sign from start to end? (b) The sign then falls to the foot of the tower. What is the magnitude of the displacement of the sign from start to this new end?

Consider \(\vec{a}\) in the positive direction of \(x, \vec{b}\) in the positive direction of \(y\), and a scalar \(d\). What is the direction of \(\vec{b} / d\) if \(d\) is (a) positive and (b) negative? What is the magnitude of (c) \(\vec{a} \cdot \vec{b}\) and (d) \(\vec{a} \cdot \vec{b} / d ?\) What is the direction of the vector resulting from (e) \(\vec{a} \times \vec{b}\) and (f) \(\vec{b} \times \vec{a}\) ? (g) What is the magnitude of the vector product in (e)? (h) What is the magnitude of the vector product in (f)? What are (i) the magnitude and (j) the direction of \(\vec{a} \times \vec{b} / d\) if \(d\) is positive?

Let \(\hat{i}\) be directed to the east, \(\hat{\mathrm{j}}\) be directed to the north, and \(\hat{\mathrm{k}}\) be directed upward. What are the values of products (a) \(\hat{i} \cdot \hat{k},(b)\) \((-\hat{\mathrm{k}}) \cdot(-\hat{\mathrm{j}})\), and \((\mathrm{c}) \hat{\mathrm{j}} \cdot(-\hat{\mathrm{j}}) ?\) What are the directions (such as east or down of products \((\mathrm{d}) \hat{\mathrm{k}} \times \hat{\mathrm{j}},(\mathrm{e})(-\hat{\mathrm{i}}) \times(-\hat{\mathrm{j}})\), and \((\mathrm{f})(-\hat{\mathrm{k}}) \times(-\hat{\mathrm{j}}) ?\)

A woman walks \(250 \mathrm{~m}\) in the direction \(30^{\circ}\) east of north, then \(175 \mathrm{~m}\) directly east. Find (a) the magnitude and (b) the angle of her final displacement from the starting point. (c) Find the distance she walks. (d) Which is greater, that distance or the magnitude of her displacement?

A vector \(\vec{d}\) has a magnitude \(3.0 \mathrm{~m}\) and is directed south. What are (a) the magnitude and (b) the direction of the vector \(5.0 \vec{d}\) ? What are (c) the magnitude and (d) the direction of the vector \(-2.0 \vec{d}\) ?

A fire ant, searching for hot sauce in a picnic area, goes through three displacements along level ground: \(\vec{d}_{1}\) for \(0.40 \mathrm{~m}\) southwest (that is, at \(45^{\circ}\) from directly south and from directly west), \(\vec{d}_{2}\) for \(0.50 \mathrm{~m}\) due east, \(\vec{d}_{3}\) for \(0.60 \mathrm{~m}\) at \(60^{\circ}\) north of east. Let the positive \(x\) direction be east and the positive \(y\) direction be north. What are (a) the \(x\) component and (b) the \(y\) component of \(\vec{d}_{1} ?\) Next, what are (c) the \(x\) component and (d) the \(y\) component of \(\vec{d}_{2}\) ? Also, what are (e) the \(x\) component and (f) the \(y\) component of \(\vec{d}_{3}\) ?

Vector \(\vec{a}\) lies in the \(y z\) plane \(63.0^{\circ}\) from the positive direction of the \(y\) axis, has a positive \(z\) component, and has magnitude \(3.20\) units. Vector \(\vec{b}\) lies in the \(x z\) plane \(48.0^{\circ}\) from the positive direction of the \(x\) axis, has a positive \(z\) component, and has magnitude \(1.40\) units. Find (a) \(\vec{a} \cdot \vec{b},(\) b) \(\vec{a} \times \vec{b}\), and (c) the angle between \(\vec{a}\) and \(\vec{b}\).

A vector \(\vec{B}\), with a magnitude of \(8.0 \mathrm{~m}\), is added to a vector \(\vec{A}\), which lies along an \(x\) axis. The sum of these two vectors is a third vector that lies along the \(y\) axis and has a magnitude that is twice the magnitude of \(\vec{A}\). What is the magnitude of \(\vec{A}\) ?

A man goes for a walk, starting from the origin of an \(x y z\) coordinate system, with the \(x y\) plane horizontal and the \(x\) axis eastward. Carrying a bad penny, he walks \(1300 \mathrm{~m}\) east, \(2200 \mathrm{~m}\) north, and then drops the penny from a cliff \(410 \mathrm{~m}\) high. (a) In unit-vector notation, what is the displacement of the penny from start to its landing point? (b) When the man returns to the origin, what is the magnitude of his displacement for the return trip?