Mathematics - MCQ Practice Questions

Understanding:

We must compute the dot product of two vectors.

Formula:

Step 1: Multiply corresponding components

Answer:

The dot product $\vec{a}\cdot \vec{b}$ equals $-8$.

Quick Tip:

A common error is to forget the sign of the third component; always carry the minus sign through each multiplication carefully.

Understanding:

We must find $|\vec{a}\times \vec{b}|$.

Formula:

Step 1: Evaluate the determinant

Step 2: Find the magnitude

Answer:

The magnitude of $\vec{a}\times \vec{b}$ is $\sqrt{3}$.

Quick Tip:

$|\vec{a}\times \vec{b}|$ gives the area of the parallelogram formed by $\vec{a}$ and $\vec{b}$; here that area is $\sqrt{3}$.

Understanding:

We must find the angle $\theta$ between two vectors using the dot product formula.

Formula:

Step 1: Compute the dot product

Step 2: Compute the magnitudes

Step 3: Find $\cos \theta$

Step 4: Find $\theta$

Answer:

The angle between the vectors is ${\cos }^{-1}\left(\frac{1}{3}\right)$.

Understanding:

We must find the unit vector along $\vec{v}=3\hat{i}-4\hat{j}$.

Formula:

Step 1: Compute the magnitude

Step 2: Divide by the magnitude

Answer:

The unit vector in the direction of $\vec{v}$ is $\frac{3}{5}\hat{i}-\frac{4}{5}\hat{j}$.

Quick Tip:

A common trap is preserving the sign: the $j$-component is negative, matching the original vector's direction.

Understanding:

We must find the distance between points $A$ and $B$ given their position vectors.

Formula:

Step 1: Compute $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$

Step 2: Find the magnitude

Answer:

The length of $AB$ is $2\sqrt{6}$.

Understanding:

We must evaluate the scalar triple product (box product) $[\vec{a}\vec{b}\vec{c}]$.

Formula:

Step 1: Set up the determinant

Step 2: Evaluate

Answer:

The scalar triple product equals $1$.

Quick Tip:

For the standard basis vectors $\hat{i},\hat{j},\hat{k}$ in order, the scalar triple product is always $1$; reversing any two gives $-1$.

Understanding:

We must find the direction cosines of line $OP$.

Formula:

Step 1: Find $r=|OP|$

Step 2: Compute direction cosines

Verification:

Answer:

The direction cosines are $\frac{1}{3},\frac{2}{3},\frac{2}{3}$.

Quick Tip:

Always verify: the sum of squares of direction cosines must equal $1$.

Understanding:

We must find $\theta$ between $\vec{a}$ and $\vec{b}$.

Formula:

Step 1: Substitute values

Step 2: Find $\theta$

Answer:

The angle between the vectors is $\frac{\pi }{3}$.

Understanding:

We must find the area of triangle $ABC$ using the cross product formula.

Formula:

Step 1: Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$

Step 2: Compute $\overrightarrow{AB}\times \overrightarrow{AC}$

Step 3: Find the magnitude

Step 4: Compute area

Answer:

The area of the triangle is $\frac{\sqrt{21}}{2}$.

Quick Tip:

Remember to halve $|\overrightarrow{AB}\times \overrightarrow{AC}|$; omitting the factor of $\frac{1}{2}$ is the most common error here.

Understanding:

We need the scalar projection of $\vec{b}$ onto $\vec{a}$.

Formula:

Step 1: Compute the dot product.

Step 2: Compute $|\vec{a}|$.

Step 3: Find the projection.

Answer:

The projection of $\vec{b}$ onto $\vec{a}$ is $1$.

Quick Tip:

Projection of $\vec{b}$ on $\vec{a}$ and projection of $\vec{a}$ on $\vec{b}$ are generally different — don't confuse the two.

Understanding:

Three unit vectors sum to zero; find the sum of their pairwise dot products.

Formula:

Square both sides of the vector equation:

Expanding:

Step 1: Substitute the magnitudes.

Step 2: Solve.

Answer:

The value of $\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ is $-\frac{3}{2}$.

Quick Tip:

This identity — squaring a sum of vectors — is one of the most reused tricks in vector theory. Memorise the expansion $|\vec{a}+\vec{b}+\vec{c}{|}^2=|\vec{a}{|}^2+|\vec{b}{|}^2+|\vec{c}{|}^2+2(\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a})$.

Understanding:

Find the area of a triangle given three vertices in 3D.

Formula:

Step 1: Find the edge vectors.

Step 2: Compute the cross product.

Step 3: Find the magnitude.

Step 4: Compute the area.

Answer:

The area of the triangle is $\frac{3\sqrt{3}}{2}$ square units.

Quick Tip:

If all three components of $\overrightarrow{AB}\times \overrightarrow{AC}$ are equal, the triangle is equilateral — a useful symmetry check.

Understanding:

Find the perpendicular distance between two parallel planes.

Formula:

First, write both planes in the form $ax+by+cz+d=0$ with the same normal direction, then use:

Step 1: Make the coefficients of Plane 2 match Plane 1 by dividing by 2.

So $d_1=4$ and $d_2=\frac{7}{2}$.

Step 2: Compute the denominator.

Step 3: Compute the distance.

Answer:

The distance between the two parallel planes is $\frac{1}{2\sqrt{14}}$.

Quick Tip:

Always reduce both planes to the same coefficient scale before applying the distance formula — the most common mistake is using $|d_1-d_2|=|4-7|=3$ without halving, giving a wrong answer.

Understanding:

Find $\lambda$ such that $\vec{a}\perp \vec{b}$.

Formula:

Two vectors are perpendicular if and only if their dot product is zero:

Step 1: Expand the dot product.

Step 2: Set equal to zero and solve.

Answer:

The value of $\lambda$ is $-\frac{1}{3}$.

Understanding:

Find the foot of perpendicular from $P(1,3,4)$ to the plane $2x-y+z+3=0$.

Formula:

The foot $Q$ lies on the line through $P$ in the direction of the normal $\vec{n}=(2,-1,1)$:

Substitute $Q$ into the plane equation to find $t$.

Step 1: Substitute $Q$ into $2x-y+z+3=0$.

Step 2: Find the coordinates of the foot.

Verification:

Check $(-1,4,3)$ lies on the plane:

Answer:

The foot of the perpendicular is $(-1,4,3)$.

Quick Tip:

Always verify the foot by substituting back into the plane equation — it catches sign errors instantly.

Understanding:

Find the volume of the parallelepiped with edge vectors:

Formula:

Step 1: Expand along the first row.

Step 2: Evaluate the $2\times 2$ determinants.

Step 3: Combine.

Answer:

The volume of the parallelepiped is $10$ cubic units.

Quick Tip:

If the scalar triple product equals zero, the three vectors are coplanar — a key fact tested separately in many exams.

Understanding:

Find the shortest distance between two skew lines.

Formula:

Step 1: $\overrightarrow{a_2}-\overrightarrow{a_1}$.

Step 2: $\overrightarrow{d_1}\times \overrightarrow{d_2}$.

Step 3: Dot product of $\overrightarrow{a_2}-\overrightarrow{a_1}$ and $\overrightarrow{d_1}\times \overrightarrow{d_2}$.

Step 4: $|\overrightarrow{d_1}\times \overrightarrow{d_2}|$.

Step 5: Shortest distance.

Answer:

The shortest distance between the two lines is $\frac{3\sqrt{2}}{2}$.

Quick Tip:

If $\overrightarrow{d_1}\times \overrightarrow{d_2}=\vec{0}$, the lines are parallel and a different formula applies. Always check the cross product first.

Understanding:

Find the unit vector in the direction of $\vec{a}+\vec{b}+\vec{c}$.

Formula:

Step 1: Compute the sum.

Step 2: Find the magnitude.

Step 3: Compute the unit vector.

Answer:

The unit vector in the direction of $\vec{a}+\vec{b}+\vec{c}$ is $\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$.

Quick Tip:

The unit vector along $\hat{i}+\hat{j}+\hat{k}$ has each component $\frac{1}{\sqrt{3}}$ — this appears frequently in problems involving body diagonals of a cube.

Understanding:

Find the plane through $(1,-1,2)$ perpendicular to the planes $2x+3y-2z=5$ and $x+2y-3z=8$.

Formula:

The normal to the required plane is perpendicular to the normals of both given planes, so it equals their cross product:

where $\overrightarrow{n_1}=(2,3,-2)$ and $\overrightarrow{n_2}=(1,2,-3)$.

Step 1: Compute $\overrightarrow{n_1}\times \overrightarrow{n_2}$.