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

$$\vec{u} \times (\vec{v} + \vec{w}) = \vec{u} \times \vec{v} + \vec{u} \times \vec{w}$$

Q: Which of the following is the correct determinant formula for $$\vec{u} \times \vec{v} if$$ $$\vec{u} = (u_1$, u_2$, u_3$)$$ and $$\vec{v} = (v_1$, v_2$, v_3$)$$?

$$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1$ & u_2$ & u_3$ \\ v_1$ & v_2$ & v_3$ \end{vmatrix}$$

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

$$\vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \times \vec{v}) \times \vec{w}$$

Question 1

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

Accepted Answer

It is perpendicular to both $$\vec{u}$$ and $$\vec{v}.$$

Question 2

Given $$\vec{u} = 2\hat{i} + 3\hat{j} + \hat{k}$$ and $$\vec{v} = \hat{i} - 2\hat{j} + 4\hat{k}$$, what is the $$\hat{j}$$ component of $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

$$7$$

Question 3

If $$\vec{u}$$ and $$\vec{v}$$ are parallel vectors, what is $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

$$\vec{0}$$

Question 4

Which formula correctly gives the area of the parallelogram formed by vectors $$\vec{u}$$ and $$\vec{v}$$?

Accepted Answer

$$|\vec{u} 	imes \vec{v}|$$

Question 5

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

Accepted Answer

$$\vec{u} 	imes (\vec{v} + \vec{w}) = \vec{u} 	imes \vec{v} + \vec{u} 	imes \vec{w}$$

Question 6

Given $$\vec{u} = 3\hat{i} - \hat{j} + 2\hat{k}$$ and $$\vec{v} = \hat{i} + \hat{j} - 3\hat{k}$$, compute $$\vec{u} 	imes \vec{v}.$$

Accepted Answer

$$\hat{i} + 11\hat{j} + 4\hat{k}$$

Question 7

The scalar triple product $$\vec{u} \cdot (\vec{v} 	imes \vec{w})$$ gives:

Accepted Answer

The volume of the parallelepiped formed by $$\vec{u}$$, $$\vec{v}$$, and $$\vec{w}$$

Question 8

Which of the following is the correct determinant formula for $$\vec{u} 	imes \vec{v} if$$ $$\vec{u} = (u_1$, u_2$, u_3$)$$ and $$\vec{v} = (v_1$, v_2$, v_3$)$$?

Accepted Answer

$$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ u_1$ & u_2$ & u_3$ \ v_1$ & v_2$ & v_3$ \end{vmatrix}$$

Question 9

If $$\vec{u} = u_1$\hat{i} + u_2$\hat{j} + u_3$\hat{k}$$ and $$\vec{v} = v_1$\hat{i} + v_2$\hat{j} + v_3$\hat{k}$$, what is the $$\hat{k}$$ component of $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

$$(u_1v_2 - u_2v_1)$$

Question 10

A force $$\vec{F} is $$applied at a point with position vector $$\vec{r}. $$The torque $$\vec{	au}$$ about the origin is given by:

Accepted Answer

$$\vec{r} 	imes \vec{F}$$

Question 11

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

Accepted Answer

$$\vec{u} 	imes (\vec{v} 	imes \vec{w}) = (\vec{u} 	imes \vec{v}) 	imes \vec{w}$$

Question 12

If $$\vec{u} = 2\hat{i} + 3\hat{j}$$ and $$\vec{v} = 4\hat{i} + 6\hat{j}$$, what is $$\vec{u} 	imes \vec{v}$$?

Accepted Answer

$$\vec{0}$$

Question 13

The right-hand rule for the cross product $$\vec{u} 	imes \vec{v}$$ determines:

Accepted Answer

The direction of the cross product

Question 14

Which of the following physical quantities can be represented as a cross product?

Accepted Answer

Torque

Question 15

If $$\vec{u} \cdot (\vec{v} 	imes \vec{w}) = 0$$, what can be concluded about the vectors?

Accepted Answer

They are coplanar

Question 16

Given $$\vec{u} = \hat{i}$$, $$\vec{v} = \hat{j}$$, and $$\vec{w} = \hat{k}$$, what is $$\vec{u} \cdot (\vec{v} 	imes \vec{w})$$?

Accepted Answer

$$1$$

Question 17

A charged particle with charge $$q$$ moves with velocity $$\vec{v} in a $$magnetic field $$\vec{B}. $$The electromagnetic force $$\vec{F} on $$the particle is:

Accepted Answer

$$q(\vec{v} 	imes \vec{B})$$

The Cross Product and Its Applications in 3D Calculus

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