An airplane starts from an airport located at the origin O and flies 150 miles in the

In Problems 1–8, find (a) 3a, (b) a+b, (c) a-b, (d) ||a+b||, and (e) ||a - b||. \(\mathbf{a}=2 \mathbf{i}+4 \mathbf{j}, \mathbf{b}=-\mathbf{i}+4 \mathbf{j}\)

In Problems 1–8, find (a) 3a, (b) a+b, (c) a-b, (d) ||a+b||, and (e) ||a - b||. \(\mathbf{a}=\langle 1,1\rangle, \mathbf{b}=\langle 2,3\rangle\)

In Problems 1–8, find (a) 3a, (b) a+b, (c) a-b, (d) ||a+b||, and (e) ||a - b||. \(\mathbf{a}=\langle 4,0\rangle, \mathbf{b}=\langle 0,-5\rangle\)

In Problems 1–8, find (a) 3a, (b) a+b, (c) a-b, (d) ||a+b||, and (e) ||a - b||. \(\mathbf{a}=\frac{1}{6} \mathbf{i}-\frac{1}{6} \mathbf{j}, \mathbf{b}=\frac{1}{2} \mathbf{i}+\frac{5}{6} \mathbf{j}\)

In Problems 1–8, find (a) 3a, (b) a+b, (c) a-b, (d) ||a+b||, and (e) ||a - b||. \(\mathbf{a}=-3 \mathbf{i}+2 \mathbf{j}, \mathbf{b}=7 \mathbf{j}\)

In Problems 1–8, find (a) 3a, (b) a+b, (c) a-b, (d) ||a+b||, and (e) ||a - b||. \(\mathbf{a}=\langle 1,3\rangle, \mathbf{b}=-5 \mathbf{a}\)

In Problems 1–8, find (a) 3a, (b) a+b, (c) a-b, (d) ||a+b||, and (e) ||a - b||. \(\mathbf{a}=-\mathbf{b}, \mathbf{b}=2 \mathbf{i}-9 \mathbf{j}\)

In Problems 1–8, find (a) 3a, (b) a+b, (c) a-b, (d) ||a+b||, and (e) ||a - b||. \(\mathbf{a}=\langle 7,10\rangle, \mathbf{b}=\langle 1,2\rangle\)

In Problems 9–14, find (a) 4a-2b and (b) -3a-5b. \(\mathbf{a}=\langle 1,-3\rangle, \mathbf{b}=\langle-1,1\rangle\)

In Problems 9–14, find (a) 4a-2b and (b) -3a-5b. \(\mathbf{a}=\mathbf{i}+\mathbf{j}, \mathbf{b}=3 \mathbf{i}-2 \mathbf{j}\)

In Problems 9–14, find (a) 4a-2b and (b) -3a-5b. \(\mathbf{a}=\mathbf{i}-\mathbf{j}, \mathbf{b}=-3 \mathbf{i}+4 \mathbf{j}\)

In Problems 9–14, find (a) 4a-2b and (b) -3a-5b. \(\mathbf{a}=\langle 2,0\rangle, \mathbf{b}=\langle 0,-3\rangle\)

In Problems 9–14, find (a) 4a-2b and (b) -3a-5b. \(\mathbf{a}=\langle 4,10\rangle, \mathbf{b}=-2\langle 1,3\rangle\)

In Problems 9–14, find (a) 4a-2b and (b) -3a-5b. \(\mathbf{a}=\langle 3,1\rangle+\langle-1,2\rangle, \mathbf{b}=\langle 6,5\rangle-\langle 1,2\rangle\)

In Problems 15–18, find the vector \(\overrightarrow{P_{1} P_{2}}\). Graph \(\overrightarrow{P_{1} P_{2}}\) and its corresponding position vector. \(P_{1}(3,2), P_{2}(5,7)\)

In Problems 15–18, find the vector \(\overrightarrow{P_{1} P_{2}}\). Graph \(\overrightarrow{P_{1} P_{2}}\) and its corresponding position vector. \(P_{1}(-2,-1), P_{2}(4,-5)\)

In Problems 15–18, find the vector \(\overrightarrow{P_{1} P_{2}}\). Graph \(\overrightarrow{P_{1} P_{2}}\) and its corresponding position vector. \(P_{1}(3,3), P_{2}(5,5)\)

In Problems 15–18, find the vector \(\overrightarrow{P_{1} P_{2}}\). Graph \(\overrightarrow{P_{1} P_{2}}\) and its corresponding position vector. \(P_{1}(0,3), P_{2}(2,0)\)

Find the terminal point of the vector \(\overrightarrow{P_{1} P_{2}}=4 \mathbf{i}+8 \mathbf{j}\) if its initial point is (-3, 10)

Find the initial point of the vector \(\overrightarrow{P_{1} P_{2}}=\langle-5,-1\rangle\) if its terminal point is \(\langle 4,7\rangle .\)

Determine which of the following vectors are parallel to \(\mathbf{a}=4 \mathbf{i}+6 \mathbf{j}\). (a) \(-4 \mathbf{i}-6 \mathbf{j}\) (b) \(-\mathbf{i}-\frac{3}{2} \mathbf{j}\) (c) \(10 \mathbf{i}+15 \mathbf{j}\) (d) \(2(\mathbf{i}-\mathbf{j})-3\left(\frac{1}{2} \mathbf{i}-\frac{5}{12} \mathbf{j}\right)\) (e) \(8 \mathbf{i}+12 \mathbf{j}\) (f ) \((5 \mathbf{i}+\mathbf{j})-(7 \mathbf{i}+4 \mathbf{j})\)

Determine a scalar c so that \(\mathbf{a}=3 \mathbf{i}+c \mathbf{j} \text { and } \mathbf{b}=-\mathbf{i}+9 \mathbf{j}\) are parallel.

In Problems 23 and 24, find a+(b+c) for the given vectors. \(\mathbf{a}=\langle 5,1\rangle, \mathbf{b}=\langle-2,4\rangle, \mathbf{c}=\langle 3,10\rangle\)

In Problems 23 and 24, find a+(b+c) for the given vectors. \(\mathbf{a}=\langle 1,1\rangle, \mathbf{b}=\langle 4,3\rangle, \mathbf{c}=\langle 0,-2\rangle\)

In Problems 25–28, find a unit vector (a) in the same direction as a, and (b) in the opposite direction of a. \(\mathbf{a}=\langle 2,2\rangle\)

In Problems 25–28, find a unit vector (a) in the same direction as a, and (b) in the opposite direction of a. \(\mathbf{a}=\langle-3,4\rangle\)

In Problems 25–28, find a unit vector (a) in the same direction as a, and (b) in the opposite direction of a. \(\mathbf{a}=\langle 0,-5\rangle\)

In Problems 25–28, find a unit vector (a) in the same direction as a, and (b) in the opposite direction of a. \(\mathbf{a}=\langle 1,-\sqrt{3}\rangle\)

In Problems 29 and 30, \(\mathbf{a}=\langle 2,8\rangle \text { and } \mathbf{b}=\langle 3,4\rangle\). Find a unit vector in the same direction as the given vector. a+b

In Problems 29 and 30, \(\mathbf{a}=\langle 2,8\rangle \text { and } \mathbf{b}=\langle 3,4\rangle\). Find a unit vector in the same direction as the given vector. 2a-3b

In Problems 31 and 32, find a vector b that is parallel to the given vector and has the indicated magnitude. \(\mathbf{a}=3 \mathbf{i}+7 \mathbf{j},\|\mathbf{b}\|=2\)

In Problems 31 and 32, find a vector b that is parallel to the given vector and has the indicated magnitude. \(\mathbf{a}=\frac{1}{2} \mathbf{i}-\frac{1}{2} \mathbf{j},\|\mathbf{b}\|=3\)

Find a vector in the opposite direction of \(a=\langle 4,10\rangle\) but \(\frac{3}{4}\) as long.

Given that \(\mathbf{a}=\langle 1,1\rangle \text { and } \mathbf{b}=\langle-1,0\rangle\), find a vector in the same direction as a+b but five times as long.

In Problems 35 and 36, use the given figure to draw the indicated vector. \(3 \mathbf{b}-\mathbf{a}\)

In Problems 35 and 36, use the given figure to draw the indicated vector. \(\mathbf{a}+(\mathbf{b}+\mathbf{c})\)

In Problems 37 and 38, express the vector x in terms of the vectors a and b.

In Problems 37 and 38, express the vector x in terms of the vectors a and b.

In Problems 39 and 40, use the given figure to prove the given result. a + b + c = 0

In Problems 39 and 40, use the given figure to prove the given result. a +b + c + d = 0

In Problems 41 and 42, express the vector a = 2i + 3j as a linear combination of the given vectors b and c. b = i + j, c = i - j

In Problems 41 and 42, express the vector a = 2i + 3j as a linear combination of the given vectors b and c. b = -2i + 4j, c = 5i + 7j

A vector is said to be tangent to a curve at a point if it is parallel to the tangent line at the point. In Problems 43 and 44, find a unit tangent vector to the given curve at the indicated point. \(y=\frac{1}{4} x^{2}+1,(2,2)\)

A vector is said to be tangent to a curve at a point if it is parallel to the tangent line at the point. In Problems 43 and 44, find a unit tangent vector to the given curve at the indicated point. \(y=-x^{2}+3 x,(0,0)\)

When walking, a person's foot strikes the ground with a force F at an angle \(\theta\) from the vertical. In FIGURE 7.1.18, the vector F is resolved into vector components \(\mathbf{F}_{g}\), which is parallel to the ground, and \(\mathbf{F}_{n}\) which is perpendicular to the ground. In order that the foot does not slip, the force \(\mathbf{F}_{g}\) must be offset by the opposing force \(\mathbf{F}_{f}\) of friction; that is, \(\mathbf{F}_{f}=-\mathbf{F}_{g}\). (a) Use the fact that \(\left\|\mathbf{F}_{f}\right\|=\mu\left\|\mathbf{F}_{n}\right\|\) where \(\mu\) is the coefficient of friction, to show that \(\tan \theta=\mu\). The foot will not slip for angles less than or equal to \(\theta\). (b) Given that \(\mu=0.6\) for a rubber heel striking an asphalt sidewalk, find the "no-slip" angle.

A 200-1b traffic light supported by two cables hangs in equilibrium. As shown in FIGURE 7.1.19(b), let the weight of the light be represented by w and the forces in the two cables by \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\). From Figure 7.1.19(c), we see that a condition of equilibrium is \(\mathbf{w}+\mathbf{F}_{1}+\mathbf{F}_{2}=\mathbf{0}\). See Problem 39. If \(\begin{aligned} \mathbf{w} &=-200 \mathbf{j} \\ \mathbf{F}_{1} &=\left(\left\|\mathbf{F}_{1}\right\| \cos 20^{\circ}\right) \mathbf{i}+\left(\left\|\mathbf{F}_{1}\right\| \sin 20^{\circ}\right) \mathbf{j} \\ \mathbf{F}_{2} &=-\left(\left\|\mathbf{F}_{2}\right\| \cos 15^{\circ}\right) \mathbf{i}+\left(\left\|\mathbf{F}_{2}\right\| \sin 15^{\circ}\right) \mathbf{j} \end{aligned} \) use (7) to determine the magnitudes of \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\). [Hint: Reread (iii) of Definition 7.1.1.]

An electric charge Q is uniformly distributed along the y-axis between y = -a and y = a. See FIGURE 7.1.20. The total force exerted on the charge q on the x-axis by the charge Q is \(\mathbf{F}=F_{x} \mathbf{i}+F_{y} \mathbf{j}\), where \(F_{x}=\frac{q Q}{4 \pi \varepsilon_{0}} \int_{-a}^{a} \frac{L d y}{2 a\left(L^{2}+y^{2}\right)^{3 / 2}}\) and \(F_{y}=-\frac{q Q}{4 \pi \varepsilon_{0}} \int_{-a}^{a} \frac{y d y}{2 a\left(L^{2}+y^{2}\right)^{3 / 2}}\). Determine F.

Using vectors, show that the diagonals of a parallelogram bisect each other. [Hint: Let M be the midpoint of one diagonal and N the midpoint of the other.]

Using vectors, show that the line segment between the midpoints of two sides of a triangle is parallel to the third side and half as long.

An airplane starts from an airport located at the origin O and flies 150 miles in the direction \(20^{\circ}\) north of east to city A. From A, the airplane then flies 200 miles in the direction 23 west of north to city B. From B, the airplane flies 240 miles in the direction \(10^{\circ}\) south of west to city C. Express the location of city C as a vector r as shown in FIGURE 7.1.21. Find the distance from O to C.