7.1E: Excersies

Exercise $\PageIndex{1}$

For the following exercises, consider points $P(−1,3), Q(1,5),$ and $R(−3,7)$. Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.

1) $\vec{PQ}$

Answer

$a. \vec{PQ}=⟨2,2⟩; b. \vec{PQ}=2i+2j$

2) $\vec{PR}$

3) $\vec{QP}$

Answer

$a. \vec{QP}=⟨−2,−2⟩; b. \vec{QP}=−2i−2j$

4) $\vec{RP}$

5) $\vec{PQ}+\vec{PR}$

Answer

$a. \vec{PQ}+\vec{PR}=⟨0,6⟩; b. \vec{PQ}+\vec{PR}=6j$

6) $\vec{PQ}−\vec{PR}$

7) $2\vec{PQ}−2\vec{PR}$

Answer

$a. 2\vec{PQ}→−2\vec{PR}=⟨8,−4⟩; b. 2\vec{PQ}−2\vec{PR}=8i−4j$

8) $2\vec{PQ}+\frac{1}{2}\vec{PR}$

9) The unit vector in the direction of $\vec{PQ}$

Answer

$a. ⟨\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}⟩; b. \frac{1}{\sqrt{2}}i+\frac{1}{\sqrt{2}}j$

10) The unit vector in the direction of $\vec{PR}$

11) A vector $v$ has initial point $(−1,−3)$ and terminal point $(2,1)$. Find the unit vector in the direction of $v$. Express the answer in component form.

Answer

$⟨\frac{3}{5},\frac{4}{5}⟩$

12) A vector $v$ has initial point $(−2,5)$ and terminal point $(3,−1)$. Find the unit vector in the direction of $v$. Express the answer in component form.

13) The vector $v$ has an initial point $P(1,0)$ and terminal point $Q$ that is on the y-axis and above the initial point. Find the coordinates of terminal point $Q$ such that the magnitude of the vector $v$ is $\sqrt{5}$.

Answer

$Q(0,2)$

14) The vector $v$ has an initial point $P(1,1)$ and terminal point $Q$ that is on the x-axis and left of the initial point. Find the coordinates of terminal point $Q$ such that the magnitude of the vector $v$ is $\sqrt{10}$.

Exercise $\PageIndex{2}$

For the following exercises, use the given vectors $a$ and $b$.

a. Determine the vector sum $a+b$ and express it in both the component form and by using the standard unit vectors.

b. Find the vector difference $a−b$ and express it in both the component form and by using the standard unit vectors.

c. Verify that the vectors $a, b,$ and $a+b$, and, respectively, $a, b$, and $a−b$ satisfy the triangle inequality.

d. Determine the vectors $2a, −b,$ and $2a−b.$ Express the vectors in both the component form and by using standard unit vectors.

15) $a=2i+j, b=i+3j$

Answer

$a. a+b=3i+4j, a+b=⟨3,4⟩;$ b. $a−b=i−2j, a−b=⟨1,−2⟩;$ c. Answers will vary; d. $2a=4i+2j, 2a=⟨4,2⟩, −b=−i−3j, −b=⟨−1,−3⟩, 2a−b=3i−j, 2a−b=⟨3,−1⟩$

16) $a=2i, b=−2i+2j$

17) Let $a$ be a standard-position vector with terminal point $(−2,−4)$. Let $b$ be a vector with initial point $(1,2)$ and terminal point $(−1,4)$. Find the magnitude of vector $−3a+b−4i+j.$

Answer

$15$

18) Let $a$ be a standard-position vector with terminal point at $(2,5)$. Let $b$ be a vector with initial point $(−1,3)$ and terminal point $(1,0)$. Find the magnitude of vector $a−3b+14i−14j.$

Exercise $\PageIndex{3}$

19) Let $u$ and $v$ be two nonzero vectors that are nonequivalent. Consider the vectors $a=4u+5v$ and $b=u+2v$ defined in terms of $u$ and $v$. Find the scalar $λ$ such that vectors $a+λb$ and $u−v$ are equivalent.

Answer

$λ=−3$

20) Let $u$ and $v$ be two nonzero vectors that are nonequivalent. Consider the vectors $a=2u−4v$ and $b=3u−7v$ defined in terms of $u$ and $v$. Find the scalars $α$ and $β$ such that vectors $αa+βb$ and $u−v$ are equivalent.

Exercise $\PageIndex{4}$

21) Consider the vector $a(t)=⟨cost,sint⟩$ with components that depend on a real number $t$. As the number $t$ varies, the components of $a(t)$ change as well, depending on the functions that define them.

a. Write the vectors $a(0)$ and $a(π)$ in component form.

b. Show that the magnitude $∥a(t)∥$ of vector $a(t)$ remains constant for any real number $t$.

c. As $t$ varies, show that the terminal point of vector $a(t)$ describes a circle centered at the origin of radius $1$.

Answer

$a. a(0)=⟨1,0⟩, a(π)=⟨−1,0⟩;$ b. Answers may vary; c. Answers may vary

22) Consider vector $a(x)=⟨x,\sqrt{1−x^2}⟩$ with components that depend on a real number $x∈[−1,1]$. As the number $x$ varies, the components of $a(x)$ change as well, depending on the functions that define them.

a. Write the vectors $a(0)$ and $a(1)$ in component form.

b. Show that the magnitude $∥a(x)∥$ of vector $a(x)$ remains constant for any real number $x$

c. As $x$ varies, show that the terminal point of vector $a(x)$ describes a circle centered at the origin of radius $1$.

23) Show that vectors $a(t)=⟨cost,sint⟩$ and $a(x)=⟨x,\sqrt{1−x^2}⟩$ are equivalent for $x=r$ and $t=2kπ$, where $k$ is an integer.

Answer

Answers may vary

24) Show that vectors $a(t)=⟨cost,sint⟩$ and $a(x)=⟨x,\sqrt{1−x^2}⟩$ are opposite for $x=r$ and $t=π+2kπ$, where $k$ is an integer.

Exercise $\PageIndex{5}$

For the following exercises, find vector $v$ with the given magnitude and in the same direction as vector $u$.

25) $‖v‖=7,u=⟨3,4⟩$

Answer

$v=⟨\frac{21}{5},\frac{28}{5}⟩$

26) $‖v‖=3,u=⟨−2,5⟩$

27) $‖v‖=7,u=⟨3,−5⟩$

Answer

$v=⟨\frac{21\sqrt{34}}{34},−\frac{35\sqrt{34}}{34}⟩$

28) $‖v‖=10,u=⟨2,−1⟩$

Exercise $\PageIndex{6}$

For the following exercises, find the component form of vector $u$, given its magnitude and the angle the vector makes with the positive x-axis. Give exact answers when possible.

29) $‖u‖=2, θ=30°$

Answer

$u=⟨\sqrt{3},1⟩$

30) $‖u‖=6, θ=60°$

31) $‖u‖=5, θ=\frac{π}{2}$

Answer

$u=⟨0,5⟩$

32) $‖u‖=8, θ=π$

33) $‖u‖=10, θ=\frac{5π}{6}$

Answer

$u=⟨−5\sqrt{3},5⟩$

34) $‖u‖=50, θ=\frac{3π}{4}$

Exercise $\PageIndex{7}$

For the following exercises, find the component form of vector $u$, given its magnitude and the angle the vector makes with the positive x-axis. Give exact answers when possible.

29) $‖u‖=2, θ=30°$

Answer

$u=⟨\sqrt{3},1⟩$

30) $‖u‖=6, θ=60°$

31) $‖u‖=5, θ=\frac{π}{2}$

Answer

$u=⟨0,5⟩$

32) $‖u‖=8, θ=π$

33) $‖u‖=10, θ=\frac{5π}{6}$

Answer

$u=⟨−5\sqrt{3},5⟩$

34) $‖u‖=50, θ=\frac{3π}{4}$

Exercise $\PageIndex{8}$

For the following exercises, vector $u$ is given. Find the angle $θ∈[0,2π)$ that vector $u$ makes with the positive direction of the x-axis, in a counter-clockwise direction.

35) $u=5\sqrt{2}i−5\sqrt{2}j$

Answer

$θ=\frac{7π}{4}$

36) $u=−\sqrt{3}i−j$

37) Let $a=⟨a_1,a_2⟩, b=⟨b_1,b_2⟩$, and $c=⟨c_1,c_2⟩$ be three nonzero vectors. If $a_1b_2−a_2b_1≠0$, then show there are two scalars, $α$ and $β$, such that $c=αa+βb.$

Answer

Answers may vary

Exercise $\PageIndex{9}$

38) Consider vectors $a=⟨2,−4⟩, b=⟨−1,2⟩,$ and $0$ Determine the scalars $α$ and $β$ such that $c=αa+βb$.

39) Let $P(x_0,f(x_0))$ be a fixed point on the graph of the differential function $f$ with a domain that is the set of real numbers.

a. Determine the real number $z_0$ such that point $Q(x_0+1,z_0)$ is situated on the line tangent to the graph of $f$ at point $P$.

b. Determine the unit vector $u$ with initial point $P$ and terminal point $Q$.

Answer

$a. z_0=f(x_0)+f′(x_0); b. u=\frac{1}{\sqrt{1+[f′(x_0)]^2}}⟨1,f′(x_0)⟩$

40) Consider the function $f(x)=x^4,$ where $x∈R$.

a. Determine the real number $z_0$ such that point $Q(2,z_0)$ s situated on the line tangent to the graph of $f$ at point $P(1,1)$.

b. Determine the unit vector $u$ with initial point $P$ and terminal point $Q$.

41) Consider $f$ and $g$ two functions defined on the same set of real numbers $D$. Let $a=⟨x,f(x)⟩$ and $b=⟨x,g(x)⟩$ be two vectors that describe the graphs of the functions, where $x∈D$. Show that if the graphs of the functions $f$ and $g$ do not intersect, then the vectors $a$ and $b$ are not equivalent.

42) Find $x∈R$ such that vectors $a=⟨x,sinx⟩$ and $b=⟨x,cosx⟩$ are equivalent.

43) Calculate the coordinates of point $D$ such that $ABCD$ is a parallelogram, with $A(1,1), B(2,4)$, and $C(7,4)$.

Answer

$D(6,1)$

44) Consider the points $A(2,1), B(10,6), C(13,4), and D(16,−2).$ Determine the component form of vector $\vec{AD}$.

Exercise $\PageIndex{10}$

45) The speed of an object is the magnitude of its related velocity vector. A football thrown by a quarterback has an initial speed of $70$ mph and an angle of elevation of $30°$. Determine the velocity vector in mph and express it in component form. (Round to two decimal places.)

Answer

$⟨60.62,35⟩$

46) A baseball player throws a baseball at an angle of $30°$ with the horizontal. If the initial speed of the ball is $100$ mph, find the horizontal and vertical components of the initial velocity vector of the baseball. (Round to two decimal places.)

47) A bullet is fired with an initial velocity of $1500$ ft/sec at an angle of $60°$ with the horizontal. Find the horizontal and vertical components of the velocity vector of the bullet. (Round to two decimal places.)

Answer

The horizontal and vertical components are $750$ ft/sec and $1299.04$ ft/sec, respectively.

48) [T] A 65-kg sprinter exerts a force of $798$ N at a $19°$ angle with respect to the ground on the starting block at the instant a race begins. Find the horizontal component of the force. (Round to two decimal places.)

49) [T] Two forces, a horizontal force of $45$ lb and another of $52$ lb, act on the same object. The angle between these forces is $25°$. Find the magnitude and direction angle from the positive x-axis of the resultant force that acts on the object. (Round to two decimal places.)

Answer

The magnitude of the resultant force is $94.71$ lb; the direction angle is $13.42°$.

50) [T] Two forces, a vertical force of $26$ lb and another of $45$ lb, act on the same object. The angle between these forces is $55°$. Find the magnitude and direction angle from the positive x-axis of the resultant force that acts on the object. (Round to two decimal places.)

51) [T] Three forces act on object. Two of the forces have the magnitudes $58$ N and $27$ N, and make angles $53°$ and $152°$, respectively, with the positive x-axis. Find the magnitude and the direction angle from the positive x-axis of the third force such that the resultant force acting on the object is zero. (Round to two decimal places.)

Answer

The magnitude of the third vector is $60.03$N; the direction angle is $259.38°$.

52) Three forces with magnitudes 80 lb, 120 lb, and 60 lb act on an object at angles of $45°, 60°$ and $30°$, respectively, with the positive x-axis. Find the magnitude and direction angle from the positive x-axis of the resultant force. (Round to two decimal places.)

53) [T] An airplane is flying in the direction of $43°$ east of north (also abbreviated as $N43E$) at a speed of $550$ mph. A wind with speed $25$ mph comes from the southwest at a bearing of $N15E$. What are the ground speed and new direction of the airplane?

Answer

The new ground speed of the airplane is $572.19$ mph; the new direction is $N41.82E.$

54) [T] A boat is travelling in the water at $30$ mph in a direction of $N20E$ (that is, $20°$ east of north). A strong current is moving at $15$ mph in a direction of $N45E$. What are the new speed and direction of the boat?

55) [T] A 50-lb weight is hung by a cable so that the two portions of the cable make angles of $40°$ and $53°$, respectively, with the horizontal. Find the magnitudes of the forces of tension $T_1$ and $T_2$ in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

Answer

$∥T_1∥=30.13lb, ∥T_2∥=38.35lb$

56) [T] A 62-lb weight hangs from a rope that makes the angles of $29°$ and $61°$, respectively, with the horizontal. Find the magnitudes of the forces of tension $T_1$ and $T_2$ in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

57) [T] A 1500-lb boat is parked on a ramp that makes an angle of $30°$ with the horizontal. The boat’s weight vector points downward and is a sum of two vectors: a horizontal vector $v_1$ that is parallel to the ramp and a vertical vector $v_2$ that is perpendicular to the inclined surface. The magnitudes of vectors $v_1$ and $v_2$ are the horizontal and vertical component, respectively, of the boat’s weight vector. Find the magnitudes of $v_1$ and $v_2$. (Round to the nearest integer.)

Answer

\(∥v1∥=750 lb, ∥v2∥=1299 lb\)

58) [T] An 85-lb box is at rest on a $26°$ incline. Determine the magnitude of the force parallel to the incline necessary to keep the box from sliding. (Round to the nearest integer.)

59) A guy-wire supports a pole that is $75$ ft high. One end of the wire is attached to the top of the pole and the other end is anchored to the ground $50$ ft from the base of the pole. Determine the horizontal and vertical components of the force of tension in the wire if its magnitude is $50$ lb. (Round to the nearest integer.)

Answer

The two horizontal and vertical components of the force of tension are $28$ lb and $42$ lb, respectively.

60) A telephone pole guy-wire has an angle of elevation of $35°$ with respect to the ground. The force of tension in the guy-wire is $120$ lb. Find the horizontal and vertical components of the force of tension. (Round to the nearest integer.)