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

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

Q: Which of the following is the correct formula for the cross product of $$\vec{a} = (a_1$, a_2$, a_3$)$$ and $$\vec{b} = (b_1$, b_2$, b_3$)$$?

$$\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$$

Question 1

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

Accepted Answer

$$(-10, 6, 8)$$

Question 2

Which of the following statements about the cross product is TRUE?

Accepted Answer

The cross product of two vectors is always perpendicular to both original vectors.

Question 3

If $$\vec{u} = (1, 2, 3)$$ and $$\vec{v} = (4, 5, 6)$$, what is the $$y-$$component of $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

$$12 - 4 = 8$$

Question 4

Which of the following best describes the result of the cross product $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

A vector perpendicular to both $$\vec{a}$$ and $$\vec{b}$$

Question 5

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

Accepted Answer

$$(0, 0, 1)$$

Question 6

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 7

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

Accepted Answer

$$(0, 0, -6) Nm$$

Question 8

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

Accepted Answer

$$\vec{a} 	imes \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$$

Question 9

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

Accepted Answer

$$\vec{0}$$

Question 10

Which of the following steps is NOT part of the standard method for computing the cross product using the determinant method?

Accepted Answer

Multiply corresponding components and add them together

Question 11

Given $$\vec{a} = (1, 2, 3)$$ and $$\vec{b} = (4, 5, 6)$$, what is the magnitude $$|\vec{a} 	imes \vec{b}|$$?

Accepted Answer

$$3.464$$

Question 12

If $$\vec{a} = (1, 0, 0)$$ and $$\vec{b} = (0, 2, 0)$$, what is the direction of $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

Along the positive $$z-$$axis

Question 13

Which of the following real-world applications involves the use of the cross product?

Accepted Answer

Calculating torque in physics

Question 14

If $$\vec{a} = (1, 2, 3)$$ and $$\vec{b} = (4, 5, 6)$$, what is the $$z-$$component of $$\vec{a} 	imes \vec{b}$$?

Accepted Answer

$$-3$$

Question 15

Which of the following best describes the geometric meaning of the magnitude of the cross product $$|\vec{a} 	imes \vec{b}|$$?

Accepted Answer

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

Question 16

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

Accepted Answer

$$(0, 1, 0)$$

Question 17

Which of the following is a necessary condition for the cross product of two vectors to be zero?

Accepted Answer

The vectors are parallel or one is the zero vector

Cross Product: Computation and Properties

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