Q: Two wires support a post, with dimensions as shown in the figure. If $$d = 14 m $$and the angle is $$25^\circ$$, what is the value of $$x ($$the length of the wire)?

$$x = 14 \tan 25^\circ = 6.5 m$$

Question 1

A hiker walks 1200 m due east, then turns and walks 1500 m at an angle of $$30^\circ$$ north of east. What is the hiker's total displacement from the starting point (campsite)?

Accepted Answer

$$\sqrt{(1200 + 1500 \cos 30^\$$circ)^2$$ + (1500 \sin 30^\$$circ)^2$$} = 2632 m$$

Question 2

A 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 of $$\vec{B}$$?

Accepted Answer

$$43$$ units

Question 3

A triangle has sides of length 7 cm and 11 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

$$8 cm$$

Question 4

A taxi moves five blocks due north, five blocks due east, and another three blocks due north. How far is the taxi from its starting point?

Accepted Answer

$$\sqrt{(5)^2$ + (8)^2$} = \sqrt{25 + 64} = 9$$ blocks

Question 5

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

Accepted Answer

$$35.0 \cos 325^\circ = 28.6$$ units

Question 6

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

Accepted Answer

$$(3.46, -1.0) km$$

Question 7

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 $$400 ft$$, what is its displacement vector? Let east be the $$+x$$ direction and up be $$+z.$$

Accepted Answer

$$(-1400, 0, 400) ft$$

Question 8

Vector $$\vec{A}$$ has a magnitude of $$35.0$$ units and points in the direction $$325^\circ$$ counterclockwise from the positive x-axis. Vector $$\vec{B}$$ has a magnitude of $$25.0$$ units and points in the direction $$120^\circ$$ counterclockwise from the positive x-axis. What is the y-component of $$\vec{B}$$?

Accepted Answer

$$25.0 \sin 120^\circ = 21.7$$ units

Question 9

Two wires support a post, with dimensions as shown in the figure. If $$d = 14 m $$and the angle is $$25^\circ$$, what is the value of $$x ($$the length of the wire)?

Accepted Answer

$$x = 14 	an 25^\circ = 6.5 m$$

Question 10

A vector has a magnitude of $$10$$ units and points at an angle of $$60^\circ$$ above the positive x-axis. What are its x and y components?

Accepted Answer

$$x = 10 \cos 60^\circ = 5$$, $$y = 10 \sin 60^\circ = 8.66$$

Question 11

If $$\vec{A} = (3, 4)$$ and $$\vec{B} = (-2, 1)$$, what is $$\vec{A} + \vec{B}$$?

Accepted Answer

$$(1, 5)$$

Question 12

A vector has components $$(-6, 8). $$What is its magnitude and direction (angle from the positive x-axis)?

Accepted Answer

Magnitude: $$10$$, Angle: $$126.9^\circ$$

Question 13

Which of the following is a scalar quantity?

Accepted Answer

Distance

Question 14

A person walks 3 km east, then 4 km north. What is the magnitude of their displacement?

Accepted Answer

$$\sqrt{3^2$ + 4^2$} = 5 km$$

Question 15

If $$\vec{A}$$ has a magnitude of $$5$$ units and points in the negative x-direction, and $$\vec{B}$$ has a magnitude of $$12$$ units and points in the positive y-direction, what is the magnitude of $$\vec{A} + \vec{B}$$?

Accepted Answer

$$\sqrt{5^2$ + 12^2$} = 13$$ units

Question 16

A vector $$\vec{C}$$ has components $$(7, -24). $$What is the direction of $$\vec{C}$$ measured counterclockwise from the positive x-axis?

Accepted Answer

$$-73.7^\circ or$$ $$286.3^\circ$$

Vectors and Displacement: Study Notes for Introductory Physics

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