Practice Questions

What are 'coinitial vectors'? Illustrate with a simple sketch.

State the two conditions that must be met for two vectors $\vec{a}$ and $\vec{b}$ to be considered equal.

Define a 'zero vector' and state one of its properties regarding direction.

Calculate the projection of the vector $\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}$ on the vector $\vec{b} = 2\hat{i} + 2\hat{j} + \hat{k}$.

If $\vec{a} = \hat{i} - 2\hat{j}$ and $\vec{b} = 3\hat{i} + \hat{j}$, calculate the magnitude of the vector $\vec{a} + \vec{b}$.

Formulate a condition using the dot product for three non-zero vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ to represent the sides of a right-angled triangle.

Derive the formula for the projection of vector $\vec{a}$ onto a non-zero vector $\vec{b}$ and justify each step of the derivation.

State the formula for the scalar product of two vectors $\vec{a}$ and $\vec{b}$ given in their component forms, $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$.

Calculate the unit vector in the direction of the vector $\vec{v} = 4\hat{i} - 3\hat{j} + 12\hat{k}$.

Explain the difference between a scalar and a vector quantity. Provide two examples of each from physics.

Analyze if the points A(2, -1, 1), B(1, -3, -5), and C(3, -4, -4) form the vertices of a right-angled triangle.

Recall the condition for two vectors $\vec{a}$ and $\vec{b}$ to be collinear in terms of a scalar multiple.

Recall the formula for the section formula for internal division. Find the position vector of a point R which divides the line joining two points P and Q, with position vectors $\vec{p}$ and $\vec{q}$ respectively, in the ratio 3:2 internally.

Calculate the scalar product of the vectors $\vec{a} = 5\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} - \hat{k}$.

Calculate the vector product $\vec{a} \times \vec{b}$ for $\vec{a} = 2\hat{i} + \hat{j}$ and $\vec{b} = 3\hat{j} - \hat{k}$.

Analyze if the vectors $\vec{p} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{q} = -6\hat{i} + 9\hat{j} - 3\hat{k}$ are collinear.

Justify whether the condition for equality holds in the triangle inequality, i.e., $|\vec{a} + \vec{b}| = |\vec{a}| + |\vec{b}|$, for two non-zero vectors.

Propose a method using vector products to determine if four points with position vectors $\vec{a}$, $\vec{b}$, $\vec{c}$, and $\vec{d}$ are coplanar.

The diagonals of a parallelogram are given by vectors $\vec{d_1} = 2\hat{i} - 3\hat{j} + 5\hat{k}$ and $\vec{d_2} = 4\hat{i} + \hat{j} - \hat{k}$. Formulate the vectors representing the adjacent sides of the parallelogram and then evaluate its area.

Prove that for any triangle ABC, the area can be formulated as $\frac{1}{2} |\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|$, where $\vec{a}$, $\vec{b}$, and $\vec{c}$ are the position vectors of the vertices A, B, and C respectively.

Define 'direction cosines' of a vector. If a vector has direction ratios $a, b, c$ and magnitude $r$, state the relationship between the direction cosines ($l, m, n$) and these quantities.

Describe the Triangle Law of Vector Addition. Use a diagram to explain how the sum of two vectors is represented. What is the resultant vector when the three sides of a triangle are taken in order?

Describe the concept of a 'position vector' of a point P(x, y, z) in a three-dimensional Cartesian coordinate system. Also, write the formula for its magnitude.

List and describe five fundamental types of vectors: Zero Vector, Unit Vector, Collinear Vectors, Equal Vectors, and Negative of a Vector.

Calculate the area of the triangle with vertices A(1, 2, 0), B(2, 0, 2), and C(3, 3, 1).

The diagonals of a parallelogram are given by the vectors $\vec{d_1} = 3\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{d_2} = \hat{i} - 3\hat{j} + 4\hat{k}$. Calculate the area of the parallelogram.

The position vectors of two points P and Q are $2\hat{i} + \hat{j} - 3\hat{k}$ and $3\hat{i} - 2\hat{j} + \hat{k}$ respectively. Calculate the position vector of a point R which divides the line segment PQ internally in the ratio 1:3.

Calculate the angle between the vectors $\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + 2\hat{k}$.

A student claims that if $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, then vectors $\vec{a}$ and $\vec{b}$ must be zero vectors. Critique this statement and provide the correct conclusion.

Design a non-zero vector $\vec{d}$ that is perpendicular to both $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}$, and satisfies the condition $\vec{c} \cdot \vec{d} = 1$, where $\vec{c} = \hat{i} + 2\hat{j} + 3\hat{k}$.

In a regular hexagon ABCDEF, if $\overrightarrow{AB} = \vec{a}$ and $\overrightarrow{BC} = \vec{b}$, derive the vectors for the remaining sides $\overrightarrow{CD}$, $\overrightarrow{DE}$, $\overrightarrow{EF}$, and $\overrightarrow{FA}$ in terms of $\vec{a}$ and $\vec{b}$.

For any two vectors $\vec{a}$ and $\vec{b}$, prove the Lagrange's identity: $|\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2$.

Explain the concept of scalar multiplication of a vector. Describe how multiplying a vector $\vec{a}$ by a scalar $\lambda$ affects its magnitude and direction for the cases when (i) $\lambda > 0$, (ii) $\lambda < 0$, and (iii) $\lambda = 0$.

Define the scalar (dot) product and the vector (cross) product of two vectors $\vec{a}$ and $\vec{b}$. Summarize the key differences between the two products in terms of the nature of the result and the geometric condition under which the result is zero.

Justify that the shortest distance between two skew lines, with vector equations $\vec{r} = \vec{a_1} + \lambda \vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu \vec{b_2}$, is the projection of the vector joining a point on each line onto the vector normal to both lines. Formulate the final expression for this distance.

Given the vectors $\vec{\alpha} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{\beta} = 4\hat{i} + 4\hat{j} + 7\hat{k}$. Express the vector $\vec{\beta}$ in the form $\vec{\beta} = \vec{\beta_1} + \vec{\beta_2}$, where $\vec{\beta_1}$ is parallel to $\vec{\alpha}$ and $\vec{\beta_2}$ is perpendicular to $\vec{\alpha}$.

Create a proof to show that the medians of a triangle are concurrent. Let the vertices of the triangle be A, B, and C with position vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ respectively.

If $\vec{a}$, $\vec{b}$, and $\vec{c}$ are three unit vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$, justify that $\vec{a} = \pm \frac{2}{\sqrt{3}}(\vec{b} \times \vec{c})$.

Evaluate the expression $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ for any vector $\vec{a}$. Justify your result.

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, $|\vec{a}|=5$, $|\vec{b}|=3$, and $|\vec{c}|=7$, calculate the value of $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$.

Summarize the properties of the vector (cross) product with respect to commutativity and collinearity.

A student argues that if $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ and $\vec{a} \neq \vec{0}$, then it must be true that $\vec{b} = \vec{c}$. Critique this argument and formulate the correct conclusion.

Calculate a vector of magnitude 10 units that is parallel to the resultant of vectors $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} - 3\hat{j} - \hat{k}$.

Calculate a vector $\vec{d}$ which is perpendicular to both vectors $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$ and $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$, and satisfies the condition $\vec{c} \cdot \vec{d} = 18$, where $\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$.