Q: Which of the following is NOT a property of the cross product?

$$\vec{a} \times \vec{b} = \vec{b} \times \vec{a}$$

Q: If $$\vec{a}$$ and $$\vec{b}$$ are vectors in $$\mathbb{R}^3$$, which of the following is always true?

$$\vec{a} \times \vec{b} is a $$vector in $$\mathbb{R}^3$$

Q: Which of the following is a valid distributive property of the cross product?

$$\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$$

Question 1

Given vectors $$\vec{a} = \langle 2, -1, 3 angle$$ and $$\vec{b} = \langle 0, 4, -2 angle$$, what is $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

Question 2

Which of the following statements about the cross product $$\vec{a} 	imes \vec{b} is $$TRUE?

Accepted Answer

It is perpendicular to both $$\vec{a}$$ and $$\vec{b}.$$

Question 3

If $$\vec{a} = \langle 1, 2, 3 angle$$ and $$\vec{b} = \langle 4, 5, 6 angle$$, what is the magnitude of $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

$$\sqrt{54}$$

Question 4

The area of a parallelogram with adjacent sides $$\vec{a}$$ and $$\vec{b} is $$given by:

Accepted Answer

$$|\vec{a} 	imes \vec{b}|$$

Question 5

Which property of the cross product is illustrated by $$\vec{a} 	imes \vec{b} = - (\vec{b} 	imes \vec{a})$$?

Accepted Answer

Anticommutative property

Question 6

If $$\vec{a}$$ and $$\vec{b}$$ are parallel nonzero vectors, what is $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

The zero vector

Question 7

Which of the following is the correct formula for the cross product $$\vec{a} 	imes \vec{b}$$, where $$\vec{a} = \langle a_1$, a_2$, a_3$ angle$$ and $$\vec{b} = \langle b_1$, b_2$, b_3$ angle$$?

Accepted Answer

$$\langle a_2$ b_3$ - a_3$ b_2$, a_3$ b_1$ - a_1$ b_3$, a_1$ b_2$ - a_2$ b_1$ \rangle$$

Question 8

A force $$\vec{F} = \langle 3, 0, 0 angle N $$acts at a point with position vector $$\vec{r} = \langle 0, 2, 0 angle m. $$What is the torque $$\vec{	au} = \vec{r} 	imes \vec{F}$$?

Accepted Answer

$$\langle 0, 0, 6 \rangle Nm$$

Question 9

If $$\vec{a} = \langle 1, 0, 0 angle$$ and $$\vec{b} = \langle 0, 1, 0 angle$$, what is the direction of $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

Positive $$z-$$axis

Question 10

Which of the following is NOT a property of the cross product?

Accepted Answer

$$\vec{a} 	imes \vec{b} = \vec{b} 	imes \vec{a}$$

Question 11

Given $$\vec{a} = \langle 1, 2, 3 angle$$ and $$\vec{b} = \langle 4, 5, 6 angle$$, compute $$\vec{a} 	imes \vec{b}.$$

Accepted Answer

$$\langle -3, 6, -3 \rangle$$

Question 12

If $$\vec{a}$$ and $$\vec{b}$$ are vectors in $$\mathbb{R}^3$$, which of the following is always true?

Accepted Answer

$$\vec{a} 	imes \vec{b} is a $$vector in $$\mathbb{R}^3$$

Question 13

The cross product $$\vec{a} 	imes \vec{b}$$ can be interpreted geometrically as:

Accepted Answer

The area of the parallelogram formed by $$\vec{a}$$ and $$\vec{b}$$

Question 14

If $$\vec{a} = \langle 0, 0, 1 angle$$ and $$\vec{b} = \langle 1, 0, 0 angle$$, what is $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

$$\langle 0, 1, 0 \rangle$$

Question 15

Which of the following best describes the right-hand rule for the cross product?

Accepted Answer

Point your right fingers in the direction of $$\vec{a}$$, curl them toward $$\vec{b}$$, and your thumb points in the direction of $$\vec{a} 	imes \vec{b}.$$

Question 16

If $$\vec{a} = \langle 1, 2, 3 angle$$ and $$\vec{b} = \langle 4, 5, 6 angle$$, what is the value of $$\vec{a} \cdot (\vec{a} 	imes \vec{b})$$?

Accepted Answer

$$0$$

Question 17

Which of the following is a valid distributive property of the cross product?

Accepted Answer

$$\vec{a} 	imes (\vec{b} + \vec{c}) = \vec{a} 	imes \vec{b} + \vec{a} 	imes \vec{c}$$

Question 18

A parallelogram has vertices $$A(1,2,3)$$, $$B(4,5,6)$$, $$C(7,8,9)$$, and $$D(10,11,12). $$What is the area of the parallelogram?

Accepted Answer

$$0$$

Section 2.4 - The Cross Product

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