Q: A hiker walks 8 km west and then 6 km south. What is the direction (angle south of west) of the displacement vector?

$$\tan$$^{-1}$$\left(\frac{6}{8}\right) \approx 37^\circ$$ south of west

Q: A boat travels 4 km north and then 3 km east. What is the magnitude and direction of the displacement vector?

Magnitude: $$5 km$$, Direction: $$\tan$$^{-1}$$\left(\frac{3}{4}\right) \approx 37^\circ$$ east of north

Question 1

Given vectors $$\vec{u}$$ and $$\vec{v} in $$the plane, which of the following correctly represents the vector sum $$\vec{u} + \vec{v}$$ using the tip-to-tail method?

Accepted Answer

Place the tail of $$\vec{v} at $$the tip of $$\vec{u}$$ and draw a vector from the tail of $$\vec{u} to $$the tip of $$\vec{v}.$$

Question 2

If $$\vec{u} = \langle 3, 4 \rangle$$ and $$\vec{v} = \langle -1, 2 \rangle$$, what is $$\vec{u} + \vec{v}$$?

Accepted Answer

$$\langle 2, 6 \rangle$$

Question 3

Which property of vector addition is illustrated when $$\vec{u} + \vec{v} = \vec{v} + \vec{u}$$?

Accepted Answer

Commutative property

Question 4

A pilot flies 5 km east, then 12 km north. What is the magnitude of the displacement vector?

Accepted Answer

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

Question 5

If $$\vec{a} = \langle 2, -3 \rangle$$ and $$\vec{b} = \langle -2, 3 \rangle$$, what is $$\vec{a} + \vec{b}$$?

Accepted Answer

$$\langle 0, 0 \rangle$$

Question 6

Which of the following best describes the geometric representation of vector subtraction $$\vec{u} - \vec{v}$$?

Accepted Answer

Draw $$\vec{u}$$ and then $$-\vec{v}$$ from the tip of $$\vec{u}.$$

Question 7

A hiker walks 8 km west and then 6 km south. What is the direction (angle south of west) of the displacement vector?

Accepted Answer

$$	an$$^{-1}$$\left(\frac{6}{8}ight) \approx 37^\circ$$ south of west

Question 8

If $$\vec{u} = \langle 7, -2 \rangle$$ and $$\vec{v} = \langle 3, 5 \rangle$$, what is the magnitude of $$\vec{u} + \vec{v}$$?

Accepted Answer

$$\sqrt{(7+3)^2$ + (-2+5)^2$} = \sqrt{10^2$ + 3^2$} = \sqrt{109}$$

Question 9

Which of the following is NOT a property of vector addition?

Accepted Answer

Multiplicative property

Question 10

A boat travels 4 km north and then 3 km east. What is the magnitude and direction of the displacement vector?

Accepted Answer

Magnitude: $$5 km$$, Direction: $$	an$$^{-1}$$\left(\frac{3}{4}ight) \approx 37^\circ$$ east of north

Question 11

If $$\vec{u} = \langle 2, 5 \rangle$$ and $$\vec{v} = \langle -4, 1 \rangle$$, what is $$\vec{u} - \vec{v}$$?

Accepted Answer

Question 12

Which scenario best illustrates the use of vectors in real life?

Accepted Answer

Determining the net force acting on an object from multiple directions

Question 13

If $$\vec{u} = \langle 0, 7 \rangle$$ and $$\vec{v} = \langle 5, 0 \rangle$$, what is the magnitude of $$\vec{u} + \vec{v}$$?

Accepted Answer

$$\sqrt{5^2$ + 7^2$} = \sqrt{74}$$

Question 14

A rescue helicopter flies 10 km east and then 24 km north. What is the straight-line distance from the starting point?

Accepted Answer

$$\sqrt{10^2$ + 24^2$} = 26 km$$

Question 15

Which of the following is the correct formula for the magnitude of a vector $$\vec{v} = \langle a, b \rangle$$?

Accepted Answer

$$|\vec{v}| = \sqrt{a^2$ + b^2$}$$

Vectors: Magnitude, Direction, and Operations

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