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Vector Algebra: Key Concepts and Exam Applications: 17 MCQs with answers

All 17 multiple-choice questions from this kit, free and with no sign-up.

Full kit · everything Flashcards · 36 Quick Revision · 15

Warm-up · 6 questions

Q1. Which of the following operations always returns a scalar?

Beginner
A $\vec{a}\cdot\vec{b}$
B $\vec{a}\times\vec{b}$
C $\vec{a}-\vec{b}$
D $\vec{a}+\vec{b}$
Show the answer

A · $\vec{a}\cdot\vec{b}$

The dot product (scalar product) returns a scalar. Addition, subtraction, and cross product return vectors.

Q2. If $\vec{a}\cdot\vec{b}=0$ for nonzero vectors $\vec{a}$ and $\vec{b}$, what is the angle between them?

Beginner
A $180^\circ$
B $45^\circ$
C $0^\circ$
D $90^\circ$
Show the answer

D · $90^\circ$

$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta=0$ implies $\cos\theta=0$, so $\theta=90^\circ$ (perpendicular).

Q3. What is the area of a triangle whose adjacent sides are given by vectors $\vec{a}$ and $\vec{b}$?

Beginner
A $|\vec{a}\times\vec{b}|$
B $\tfrac12|\vec{a}\times\vec{b}|$
C $|\vec{a}\cdot\vec{b}|$
D $\tfrac12|\vec{a}\cdot\vec{b}|$
Show the answer

B · $\tfrac12|\vec{a}\times\vec{b}|$

The area of a triangle is half the magnitude of the cross product of its adjacent sides.

Q4. Three vectors $\vec{a},\vec{b},\vec{c}$ are coplanar if and only if:

Beginner
A $\vec{a}\cdot(\vec{b}\times\vec{c})=0$
B $\vec{a}\times(\vec{b}\times\vec{c})=\vec{0}$
C $\vec{a}\cdot\vec{b}\cdot\vec{c}=0$
D $\vec{a}+\vec{b}+\vec{c}=\vec{0}$
Show the answer

A · $\vec{a}\cdot(\vec{b}\times\vec{c})=0$

Coplanarity is equivalent to the scalar triple product being zero: $[\vec{a}\ \vec{b}\ \vec{c}]=0$.

Q5. Which of the following is NOT a vector?

Beginner
A $\vec{a}\cdot\vec{b}$
B $\vec{a}-\vec{b}$
C $\vec{a}+\vec{b}$
D $\vec{a}\times\vec{b}$
Show the answer

A · $\vec{a}\cdot\vec{b}$

The dot product is a scalar, while the others are vectors.

Q6. The dot product of two vectors is always a scalar, while the cross product is always a vector.

Beginner
A True
B False
Show the answer

A · True

The dot product of two vectors is always a scalar, while the cross product is always a vector.

Building up · 6 questions

Q7. If $|\vec{a}\cdot\vec{b}|=|\vec{a}\times\vec{b}|$, what is the acute angle between $\vec{a}$ and $\vec{b}$?

Intermediate
A $45^\circ$
B $60^\circ$
C $30^\circ$
D $90^\circ$
Show the answer

A · $45^\circ$

The condition gives $|\cos\theta|=|\sin\theta|$, so $\tan\theta=1$ for the acute angle, hence $\theta=45^\circ$.

Q8. What is the projection of $\vec{a}=\hat{i}-\hat{j}$ on $\vec{b}=\hat{i}+\hat{j}$?

Intermediate
A $\sqrt{2}$
B $0$
C $-\sqrt{2}$
D $1$
Show the answer

B · $0$

$\vec{a}\cdot\vec{b}=1-1=0$, so the projection is $0/\sqrt{2}=0$. The vectors are perpendicular.

Q9. A unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ is given by:

Intermediate
A $\dfrac{\vec{a}-\vec{b}}{|\vec{a}-\vec{b}|}$
B $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$
C $\dfrac{\vec{a}+\vec{b}}{|\vec{a}+\vec{b}|}$
D $\pm\dfrac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|}$
Show the answer

D · $\pm\dfrac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|}$

The cross product $\vec{a}\times\vec{b}$ is perpendicular to both; normalizing gives the unit vector, with two possible signs.

Q10. If $\vec{a}$ and $\vec{b}$ are parallel, what is $\vec{a}\times\vec{b}$?

Intermediate
A $|\vec{a}||\vec{b}|$
B $\vec{0}$
C $\vec{a}\cdot\vec{b}$
D $\hat{n}$
Show the answer

B · $\vec{0}$

For parallel vectors, $\sin\theta=0$, so $\vec{a}\times\vec{b}=\vec{0}$.

Q11. What is the volume of the parallelepiped with edges $\hat{i}+\hat{j}$, $\hat{j}+\hat{k}$, $\hat{k}+\hat{i}$?

Intermediate
A $2$
B $3$
C $1$
D $0$
Show the answer

A · $2$

The scalar triple product is $2$, so the volume is $|2|=2$.

Q12. If $\vec{a}=2\hat{i}+\hat{j}-3\hat{k}$ and $\vec{b}=m\hat{i}+3\hat{j}-\hat{k}$ are perpendicular, what is $m$?

Intermediate
A $-1$
B $0$
C $2$
D $1$
Show the answer

B · $0$

$\vec{a}\cdot\vec{b}=2m+3-3=2m=0$, so $m=0$.

Topper level · 5 questions

Q13. If $\vec{a},\vec{b},\vec{c}$ are unit vectors with $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, what is $\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}$?

Advanced
A $-3/2$
B $0$
C $3/2$
D $-1$
Show the answer

A · $-3/2$

Square the sum: $0 = |\vec{a}+\vec{b}+\vec{c}|^2 = 3 + 2(\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a})$, so the sum is $-3/2$.

Q14. If $\vec{a},\vec{b},\vec{c}$ are coplanar, which of the following is necessarily true?

Advanced
A $\vec{a}\times\vec{b}=\vec{0}$
B $\vec{a}+\vec{b}$, $\vec{b}+\vec{c}$, $\vec{c}+\vec{a}$ are coplanar
C $\vec{a}\cdot\vec{b}=0$
D $\vec{a}+\vec{b}+\vec{c}=\vec{0}$
Show the answer

B · $\vec{a}+\vec{b}$, $\vec{b}+\vec{c}$, $\vec{c}+\vec{a}$ are coplanar

Linear combinations of coplanar vectors remain coplanar, so the three sums are coplanar.

Q15. If $|\vec{a}|=3$, $|\vec{b}|=4$, $|\vec{c}|=4$ and each vector is perpendicular to the sum of the other two, what is $|\vec{a}+\vec{b}+\vec{c}|$?

Advanced
A $\sqrt{17}$
B $\sqrt{41}$
C $5$
D $\sqrt{33}$
Show the answer

B · $\sqrt{41}$

The conditions give $\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}=0$, etc. Solving yields $|\vec{a}+\vec{b}+\vec{c}|^2=41$, so the magnitude is $\sqrt{41}$.

Q16. What is the value of $\hat{i}\cdot(\hat{j}\times\hat{k}) + \hat{j}\cdot(\hat{i}\times\hat{k}) + \hat{k}\cdot(\hat{i}\times\hat{j})$?

Advanced
A $1$
B $3$
C $-1$
D $0$
Show the answer

A · $1$

$\hat{i}\cdot(\hat{j}\times\hat{k})=1$, $\hat{j}\cdot(\hat{i}\times\hat{k})=-1$, $\hat{k}\cdot(\hat{i}\times\hat{j})=1$; sum is $1$.

Q17. Given vectors \( \vec{a} = 2\hat{i} + 3\hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} - \hat{j} + 4\hat{k} \), find the area of the triangle formed by these vectors.

Advanced
A 10
B 5
C 2.5
D 3
Show the answer

B · 5

The area of the triangle formed by vectors \( \vec{a} \) and \( \vec{b} \) is given by \( \text{Area} = \frac{1}{2} |\vec{a} \times \vec{b}| \). First, we calculate \( \vec{a} \times \vec{b} \):…

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