Q: If $$\vec{v} = \langle 7, -24 \rangle$$, what is the direction angle $$\theta of$$ $$\vec{v}$$ measured counterclockwise from the positive $$x-$$axis?

$$\theta = \arctan\left(\frac{-24}{7}\right)$$

Question 1

Given the vector $$\vec{v} = \langle 3, 5 \rangle$$, what is the magnitude of $$\vec{v}$$?

Accepted Answer

$$\sqrt{3^2$ + 5^2$} = \sqrt{34}$$

Question 2

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

Accepted Answer

$$\langle 1, -1 \rangle$$

Question 3

A plane travels east with a velocity vector $$\vec{v} = \langle 400, 0 \rangle km/h. A $$wind blows north at $$\vec{w} = \langle 0, 100 \rangle km/h. $$What is the resultant velocity vector?

Accepted Answer

$$\langle 400, 100 \rangle$$

Question 4

Given $$\vec{a} = \langle 6, 2 \rangle$$ and $$\vec{b} = \langle -3, 7 \rangle$$, what is $$2\vec{a} - \vec{b}$$?

Accepted Answer

$$\langle 15, -3 \rangle$$

Question 5

Which of the following best describes the geometric interpretation of adding two vectors $$\vec{u}$$ and $$\vec{v} in $$the plane?

Accepted Answer

Placing the tail of $$\vec{v} at $$the head of $$\vec{u}$$ and drawing the resultant from the tail of $$\vec{u} to $$the head of $$\vec{v}$$

Question 6

If $$\vec{v} = \langle x, y \rangle$$ and $$k is a $$scalar, what is $$k\vec{v}$$?

Accepted Answer

$$\langle kx, ky \rangle$$

Question 7

A force of $$10 N $$acts at an angle of $$60^\circ$$ above the horizontal. What are the horizontal and vertical components of the force?

Accepted Answer

$$\langle 10 \cos 60^\circ, 10 \sin 60^\circ \rangle$$

Question 8

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

Accepted Answer

$$\langle 5, -5 \rangle$$

Question 9

A boat heads due north at $$8 km/h in a $$river flowing east at $$6 km/h. $$What is the boat's actual speed relative to the ground?

Accepted Answer

$$\sqrt{8^2$ + 6^2$} = 10 km/h$$

Question 10

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

Accepted Answer

$$\sqrt{a^2$ + b^2$}$$

Question 11

If $$\vec{v} = \langle 7, -24 angle$$, what is the direction angle $$	heta of$$ $$\vec{v}$$ measured counterclockwise from the positive $$x-$$axis?

Accepted Answer

$$	heta = \arctan\left(\frac{-24}{7}ight)$$

Question 12

A vector $$\vec{v}$$ has components $$\langle x, y \rangle. If$$ $$\vec{v} is $$doubled, what happens to its magnitude?

Accepted Answer

It is multiplied by $$2$$

Question 13

Which operation is NOT valid for vectors in component form?

Accepted Answer

Multiplying two vectors component-wise to get a scalar

Question 14

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

Accepted Answer

$$\langle 10, 1 \rangle$$

Question 15

A vector $$\vec{v}$$ makes an angle of $$45^\circ$$ with the positive $$x-$$axis and has a magnitude of $$10. $$What are its components?

Accepted Answer

$$\langle 10 \cos 45^\circ, 10 \sin 45^\circ \rangle$$

Vectors in Component Form: Representation and Operations

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