Cross product Knowledge-Based Practice

Questions

1. 1.

   Which of the following expressions defines a unit vector that is perpendicular to the plane containing two non-parallel vectors $\vec{u}$ and $\vec{v}$?

   • A) $\frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|}$
   • B) $\frac{\vec{u} + \vec{v}}{|\vec{u} + \vec{v}|}$
   • C) $\frac{\vec{u} \times \vec{v}}{|\vec{u}| |\vec{v}|}$
   • D) $\frac{\vec{u} \times \vec{v}}{|\vec{u} \times \vec{v}|}$
2. 2.

   For any two non-zero vectors $\vec{A}$ and $\vec{B}$, which of the following mathematical identities correctly relates the magnitude of their cross product to their dot product?

   • A) $|\vec{A} \times \vec{B}|^2 - (\vec{A} \cdot \vec{B})^2 = |\vec{A}|^2 |\vec{B}|^2$
   • B) $|\vec{A} \times \vec{B}| + (\vec{A} \cdot \vec{B}) = |\vec{A}| |\vec{B}|$
   • C) $|\vec{A} \times \vec{B}|^2 + (\vec{A} \cdot \vec{B})^2 = 0$
   • D) $|\vec{A} \times \vec{B}|^2 + (\vec{A} \cdot \vec{B})^2 = |\vec{A}|^2 |\vec{B}|^2$
3. 3.

   Which of the following vector identities is universally valid for any three-dimensional vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$?

   • A) $\vec{A} \times \vec{B} = \vec{B} \times \vec{A}$
   • B) $\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \times \vec{B}) \times \vec{C}$
   • C) $\vec{A} \times (\vec{B} + \vec{C}) = (\vec{A} \times \vec{B}) + (\vec{A} \times \vec{C})$
   • D) $\vec{A} \times (\vec{B} + \vec{C}) = (\vec{B} \times \vec{A}) + (\vec{C} \times \vec{A})$
4. 4.

   What geometric quantity is represented by the magnitude of the vector cross product, $|\vec{A} \times \vec{B}|$, for two non-parallel vectors $\vec{A}$ and $\vec{B}$ originating from the same point?

   • A) The perimeter of the parallelogram formed by the two vectors
   • B) The length of the main diagonal connecting the tips of the two vectors
   • C) The area of the parallelogram having the two vectors as adjacent sides
   • D) Half the sum of the magnitudes of the two individual vectors
5. 5.

   Under what geometric condition is the cross product of two non-zero vectors, $\vec{A} \times \vec{B}$, equal to the zero vector?

   • A) When the vectors are perpendicular to each other
   • B) When the vectors have equal magnitudes
   • C) When the vectors form an angle of $45^\circ$ with each other
   • D) When the vectors are parallel or antiparallel to each other
6. 6.

   If two vectors satisfy $\vec{A} \times \vec{B} = 4\hat{i} - 7\hat{j} + 2\hat{k}$, what is the resulting vector of the cross product $\vec{B} \times \vec{A}$?

   • A) $4\hat{i} - 7\hat{j} + 2\hat{k}$
   • B) $4\hat{i} + 7\hat{j} + 2\hat{k}$
   • C) $-4\hat{i} - 7\hat{j} + 2\hat{k}$
   • D) $-4\hat{i} + 7\hat{j} - 2\hat{k}$
7. 7.

   For any two non-parallel vectors $\vec{A}$ and $\vec{B}$ that lie entirely in the xy-plane, in which direction does their cross product $\vec{A} \times \vec{B}$ necessarily point?

   • A) In the xy-plane, along the line bisecting the angle between them
   • B) In the xy-plane, perpendicular to vector $\vec{A}$ only
   • C) Along the z-axis, perpendicular to the xy-plane
   • D) In the xz-plane, making a 45-degree angle with the z-axis
8. 8.

   Using the standard right-handed Cartesian unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$, what is the result of the vector expression $\hat{i} \times (\hat{j} \times \hat{i})$?

   • A) $-\hat{j}$
   • B) $\hat{j}$
   • C) $\hat{k}$
   • D) $-\hat{k}$
9. 9.

   If the magnitude of the cross product of two non-zero vectors $\vec{A}$ and $\vec{B}$ is equal to their scalar product, what is the angle between them?

   • A) $45^\circ$
   • B) $30^\circ$
   • C) $60^\circ$
   • D) $90^\circ$
10. 10.

    For any two non-zero vectors $\vec{A}$ and $\vec{B}$, what is the value of the scalar product $(\vec{A} \times \vec{B}) \cdot \vec{A}$?

    • A) $|\vec{A}| |\vec{B}|$
    • B) 0
    • C) $-|\vec{A}| |\vec{B}|$
    • D) $|\vec{A}|^2 |\vec{B}|$
11. 11.

    Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes of 8 units and 5 units, respectively, and the angle between them is $30^\circ$. What is the magnitude of their cross product, $|\vec{A} \times \vec{B}|$?

    • A) 20 units
    • B) $20\sqrt{3}$ units
    • C) 40 units
    • D) 10 units
12. 12.

    Given the vectors $\vec{A} = 3\hat{i} - 2\hat{j}$ and $\vec{B} = 4\hat{j} + 5\hat{k}$, what is the cross product $\vec{A} \times \vec{B}$?

    • A) $-10\hat{i} + 15\hat{j} + 12\hat{k}$
    • B) $-10\hat{i} - 15\hat{j} + 12\hat{k}$
    • C) $10\hat{i} + 15\hat{j} - 12\hat{k}$
    • D) $12\hat{i} - 15\hat{j} - 10\hat{k}$

Answers and step-by-step solutions appear after you finish the test.