Q: A vector has components $$A_x = 10$$ and $$A_y = -24. $$What is its magnitude and direction (angle from the positive $$x-$$axis)?

Magnitude $$\sqrt{10^2$ + (-24)^2$} = 26$$, angle $$\tan$$^{-1}$$(-24/10) \approx -67.4^\circ$$

Question 1

A post is supported by two wires as shown in the figure. If the horizontal distance $$d = 14\ \mathrm{m}$$ and the angle is $$25^\circ$$, what is the value of $$x ($$the length of the wire along the hypotenuse)?

Accepted Answer

$$x = \dfrac{d}{\cos 25^\circ} = 15.4\ \mathrm{m}$$

Question 2

A triangle has sides of length $$7\ \mathrm{cm}$$ and $$11\ \mathrm{cm}. If $$the triangle is a right triangle, which of the following could be the length of the third side, to the nearest centimeter?

Accepted Answer

$$8\ \mathrm{cm}$$

Question 3

Which of the following is a vector quantity?

Accepted Answer

Acceleration

Question 4

Terrence goes for a walk. He walks $$2.0\ \mathrm{km}$$ north, then turns and walks $$4.0\ \mathrm{km}$$ south-east. What is his displacement vector? Take east to be the $$+x$$ direction and north to be the $$+y$$ direction.

Accepted Answer

$$(2.83, -2.83)\ \mathrm{km}$$

Question 5

An airplane takes off due west at an angle of $$16.0^\circ$$ with the horizontal. When the airplane is at a vertical height of $$400\ \mathrm{ft}$$, what is its displacement vector? Let east be the $$+x$$ direction and up be $$+y.$$

Accepted Answer

$$(-400 \cos 16^\circ, 400 \sin 16^\circ)\ \mathrm{ft}$$

Question 6

A taxi moves five blocks due east, five blocks due south, and another three blocks due north. Assume all blocks are of equal size. How far is the taxi from its starting point?

Accepted Answer

$$\sqrt{5^2$ + 2^2$} = 5.4$$ blocks

Question 7

A hiker walks $$1200\ \mathrm{m}$$ east, then $$1500\ \mathrm{m} at$$ $$30^\circ$$ north of east. What is the hiker's total displacement from the campsite?

Accepted Answer

$$\sqrt{(1200 + 1500 \cos 30^\$$circ)^2$$ + (1500 \sin 30^\$$circ)^2$$} \approx 2550\ \mathrm{m}$$

Question 8

Given vector $$\vec{A}$$ has a magnitude of $$29$$ units and points in the positive $$y-$$direction. When vector $$\vec{B} is $$added to $$\vec{A}$$, the resultant $$\vec{A} + \vec{B}$$ points in the negative $$y-$$direction with a magnitude of $$14$$ units. What is the magnitude and direction of $$\vec{B}$$?

Accepted Answer

Magnitude $$43$$ units, direction $$-y$$

Question 9

The magnitude of vector $$\vec{A} is$$ $$35.0$$ units and points in the direction $$325^\circ$$ counterclockwise from the positive $$x-$$axis. What are the $$x$$ and $$y$$ components of $$\vec{A}$$?

Accepted Answer

$$A_x = 35.0 \cos 325^\circ$$, $$A_y = 35.0 \sin 325^\circ$$

Question 10

Vector $$\vec{A}$$ has magnitude $$35.0$$ units at $$325^\circ$$ and vector $$\vec{B}$$ has magnitude $$25.0$$ units at $$120^\circ. $$What is the magnitude of $$\vec{A} + \vec{B}$$?

Accepted Answer

$$\sqrt{(A_x + B_x)^2$ + (A_y + B_y)^2$}$$

Question 11

A vector has components $$A_x = 10$$ and $$A_y = -24. $$What is its magnitude and direction (angle from the positive $$x-$$axis)?

Accepted Answer

Magnitude $$\sqrt{10^2$ + (-24)^2$} = 26$$, angle $$	an$$^{-1}$$(-24/10) \approx -67.4^\circ$$

Question 12

A boat travels $$5.0\ \mathrm{km}$$ east and then $$12.0\ \mathrm{km}$$ north. What is the magnitude of its displacement?

Accepted Answer

$$\sqrt{5.0^2$ + 12.0^2$} = 13.0\ \mathrm{km}$$

Question 13

A vector $$\vec{C}$$ has a magnitude of $$10$$ units and points $$60^\circ$$ above the $$x-$$axis. What are its $$x$$ and $$y$$ components?

Accepted Answer

$$C_x = 10 \cos 60^\circ$$, $$C_y = 10 \sin 60^\circ$$

Question 14

A cyclist rides $$8.0\ \mathrm{km}$$ west, then $$6.0\ \mathrm{km}$$ north. What is the direction of the displacement vector from the starting point?

Accepted Answer

$$	an$$^{-1}$$(6.0/8.0) = 36.9^\circ$$ north of west

Question 15

A rescue helicopter flies $$4.0\ \mathrm{km}$$ east, then $$3.0\ \mathrm{km}$$ south, and finally $$5.0\ \mathrm{km}$$ west. What is its final displacement from the starting point?

Accepted Answer

$$(-1.0, -3.0)\ \mathrm{km}$$

Question 16

A vector $$\vec{D}$$ has components $$D_x = -6$$ and $$D_y = 8. $$What is the magnitude and direction of $$\vec{D}$$?

Accepted Answer

Magnitude $$\sqrt{(-6)^2$ + 8^2$} = 10$$, angle $$	an$$^{-1}$$(8/-6) = -53.1^\circ$$ from $$+x-$$axis

Question 17

A hiker walks $$1200\ \mathrm{m}$$ east, then $$1500\ \mathrm{m} at$$ $$30^\circ$$ north of east. What is the angle of the hiker's final position relative to the campsite, measured from the positive $$x-$$axis?

Accepted Answer

$$	an$$^{-1}$$\left(\dfrac{1500 \sin 30^\circ}{1200 + 1500 \cos 30^\circ}ight) \approx 19.1^\circ$$

Question 18

If vector $$\vec{A}$$ has magnitude $$10$$ units at $$0^\circ$$ and vector $$\vec{B}$$ has magnitude $$10$$ units at $$90^\circ$$, what is the magnitude of $$\vec{A} + \vec{B}$$?

Accepted Answer

$$\sqrt{10^2$ + 10^2$} = 14.1$$ units

Vectors and Displacement: Study Notes for Introductory Physics

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