Vector Algebra Exam Prep
Full explanation, worked examples & memory tricks in the app
🎯 Exam Preparation Kit
Vector Algebra Exam Prep
English 109 leveled MCQs Free
Shared by a Veda learner · Generated with Veda AI
📖 Smart notes
1
What you'll study, topic by topic
❓ Solved paper preview
Sample questions with model answers
Every question in the paper gets a full exam-length answer in the app — organised by marks, the way a topper writes it.
5 marks Long answer
If \(|\vec{AC}| = |\vec{BD}|\$, show that ABCD is a rectangle.
Show answer outline
Prove that angles are right angles.
The full exam-length answer is in the app.
5 marks Long answer
Show that the vectors $\vec{a} = \hat{i} − 2\hat{j} + 3\hat{k}$, $\vec{b} = −2\hat{i} + 3\hat{j} – 4\hat{k}$ and $\vec{c} = \hat{i} – 3\hat{j} + 5\hat{k}$ are coplanar.
Show answer outline
Use scalar triple product $[\vec{a}, \vec{b}, \vec{c}] = 0$.
The full exam-length answer is in the app.
5 marks Long answer
Find a unit vector perpendicular to the plane ABC where A, B, C are points (1, 1,2), (2, 3, 5) and (1,5, 5).
Show answer outline
Use cross product of vectors AB and AC.
The full exam-length answer is in the app.
5 marks Long answer
If $\vec{b} = 2\hat{i} + \hat{j} – \hat{k}$, $\vec{c} = \hat{i} + 3\hat{k}$ and $\vec{a}$ is a unit vector. Find the maximum value of scalar triple product $[\vec{a} \vec{b} \vec{c}]$.
Show answer outline
Max value is $|\vec{b}||\vec{c}|$.
The full exam-length answer is in the app.
5 marks Long answer
Verify that $[\vec{a} \vec{b} \vec{c}] = |\vec{a}||\vec{b}||\vec{c}|$.
Show answer outline
Use properties of scalar triple product.
The full exam-length answer is in the app.
5 marks Long answer
Find a vector which is perpendicular to both $\vec{AB}$ and $\vec{AC}$ having magnitude 9 units.
Show answer outline
Use $\frac{\vec{AB} \times \vec{AC}}{|\vec{AB} \times \vec{AC}|}$.
The full exam-length answer is in the app.
5 marks Long answer
Find the area of the parallelogram whose adjacent sides are the vectors \(2\hat{i} + 3\hat{j} + 2\hat{k}\) and \(\hat{i} - \hat{j}\).
Show answer outline
Area = |\vec{u} \times \vec{v}|
The full exam-length answer is in the app.
4 marks Medium answer
If \(\vec{a}, \vec{b}, \vec{c}\) are three vectors such that \(|\vec{a}| = 3\), \(|\vec{b}| = 4\), \(|\vec{c}| = 4\) and each one of them is perpendicular to the sum of other two, then find \(|\vec{a} + \vec{b} + \vec{c}|\).
Show answer outline
|\vec{a} + \vec{b} + \vec{c}| = 0
The full exam-length answer is in the app.
4 marks Medium answer
If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are unit vectors such that \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), find the value of \(\vec{a}.\vec{b} + \vec{b}.\vec{c} + \vec{c}.\vec{a}\).
Show answer outline
-\frac{1}{2}
The full exam-length answer is in the app.
3 marks Medium answer
If \(\vec{a}, \vec{b}, \vec{c}\) are coplanar vectors, prove that \(\vec{a} + \vec{b}, \vec{b} + \vec{c}, \vec{c} + \vec{a}\) are coplanar.
Show answer outline
Use the definition of coplanarity and vector addition.
The full exam-length answer is in the app.
3 marks Medium answer
Consider the vectors \(\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}\) and \(\vec{b} = \hat{i} + 4\hat{j} - \hat{k}\). Find the projection of \(\vec{a}\) on \(\vec{b}\).
Show answer outline
Projection is given by \(\frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \vec{b}\).
The full exam-length answer is in the app.
3 marks Medium answer
A diagram shows a parallelogram ABCD. If \(\vec{AB} = 3\hat{i} - \hat{j} + 2\hat{k}\) and \(\vec{AD} = \hat{i} + \hat{j} + 2\hat{k}\). Find \(\vec{AC}\) and \(\vec{DB}\).
Show answer outline
\(\vec{AC} = \vec{AB} + \vec{AD}, \vec{DB} = \vec{AB} - \vec{AD}\)
The full exam-length answer is in the app.
3 marks Medium answer
If \(\vec{a}\) and \(\vec{b}\) are adjacent sides of any parallelogram and \(\vec{c}\) and \(\vec{d}\) are diagonals, show that \([\vec{c} \times \vec{d}] = 2 \times |\vec{a} \times \vec{b}|\).
Show answer outline
Use properties of cross products and parallelogram area.
The full exam-length answer is in the app.
3 marks Medium answer
Find $\lambda$ and $\mu$ if $(2\hat{i} + 6\hat{j} + 27\hat{k}) \times (\hat{i} + \lambda\hat{j} + \mu\hat{k}) = \vec{0}$.
Show answer outline
$\lambda = 0$, $\mu = 27$.
The full exam-length answer is in the app.
3 marks Medium answer
Using vectors, find the area of the triangle with vertices (1,1,1), (1,2,3) and (2,3,1).
Show answer outline
Area = $\frac{1}{2} |\vec{AB} \times \vec{AC}|$.
The full exam-length answer is in the app.
3 marks Medium answer
Find a vector in the direction of $\vec{a} = 2\hat{i} – \sqrt{2}\hat{j} + \sqrt{3}\hat{k}$ having magnitude 9 units.
Show answer outline
$\frac{9}{\sqrt{(2^2 + (-\sqrt{2})^2 + (\sqrt{3})^2)}}\vec{a}$.
The full exam-length answer is in the app.
3 marks Medium answer
If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + \hat{j} − \hat{k}$. Find $(\vec{a} + \vec{b}). (\vec{a} – \vec{b})$.
Show answer outline
$\vec{a} + \vec{b} = 3\hat{i} + 2\hat{j} + 0\hat{k}$.
The full exam-length answer is in the app.
3 marks Medium answer
Find the area of the triangle whose adjacent sides are respectively by $\vec{a}$ and $\vec{b}$.
Show answer outline
Area = $\frac{1}{2} |\vec{a} \times \vec{b}|$.
The full exam-length answer is in the app.
3 marks Medium answer
Find the projection of the vector $\vec{a} = \hat{i} – \hat{j}$ on the vector $\vec{b} = \hat{i} + \hat{j}$.
Show answer outline
$\frac{1}{2}(\hat{i} + \hat{j})$.
The full exam-length answer is in the app.
3 marks Medium answer
Find a unit vector perpendicular to both $\vec{a} + \vec{b}$ and $\vec{a} – \vec{b}$.
Show answer outline
Use $\frac{(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})}{|\vec{a} + \vec{b}| |\vec{a} - \vec{b}|}$.
The full exam-length answer is in the app.
3 marks Medium answer
Find the angle between the vectors $\hat{i} – 2\hat{j} + 3\hat{k}$ and $3\hat{i} – 2\hat{j} + \hat{k}$.
Show answer outline
$\cos(\theta) = \frac{\vec{a} . \vec{b}}{|\vec{a}| |\vec{b}|}$.
The full exam-length answer is in the app.
3 marks Medium answer
Find the angle between $\vec{a}$ and $\vec{b}$.
Show answer outline
$\cos(\theta) = \frac{10}{|\vec{a}||\vec{b}|}$.
The full exam-length answer is in the app.
3 marks Medium answer
Consider the vectors $\vec{a} = \hat{i} – 7\hat{j} + 7\hat{k}$, $\vec{b} = 3\hat{i} – 2\hat{j} + 2\hat{k}$. Find $\vec{a} \times \vec{b}$.
Show answer outline
$\vec{a} \times \vec{b} = |\hat{i} \hat{j} \hat{k}|$.
The full exam-length answer is in the app.
3 marks Medium answer
Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} – 7\hat{j} + \hat{k}$.
Show answer outline
Area = $|\vec{a} \times \vec{b}|$.
The full exam-length answer is in the app.
3 marks Medium answer
Write a vector from the point A(1, 3, 5) to B (4, 3, 2).
Show answer outline
$3\hat{i} + 0\hat{j} - 3\hat{k}$.
The full exam-length answer is in the app.
3 marks Medium answer
Find a unit vector perpendicular to $\vec{a}$ and $\vec{b}$.
Show answer outline
Use $\frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$.
The full exam-length answer is in the app.
3 marks Medium answer
Find the projection of the vector $\hat{i} + 3\hat{j} + 7\hat{k}$ on the vector $7\hat{i} – \hat{j} + 8\hat{k}$.
Show answer outline
$\frac{\hat{i} + 3\hat{j} + 7\hat{k}}{|7\hat{i} - \hat{j} + 8\hat{k}|}$.
The full exam-length answer is in the app.
3 marks Medium answer
Consider the vectors $\vec{a} = 3\hat{i} + \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{i}− \hat{j} + \hat{k}$. Find $\vec{a} \times \vec{b}$.
Show answer outline
$\vec{a} \times \vec{b} = (3 - 0)\hat{i} + (0 - 4)\hat{j} + (3 + 1)\hat{k}$.
The full exam-length answer is in the app.
3 marks Medium answer
Consider the vectors $\vec{a} = 2\hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \hat{j} + \hat{k}$. Find a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$.
Show answer outline
Use $\frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$.
The full exam-length answer is in the app.
3 marks Medium answer
If $\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} – 2\hat{k}$, then show that $(\vec{a} – \vec{b}) \times (\vec{a} + \vec{b}) = 2(\vec{a} \times \vec{b})$.
Show answer outline
Use properties of cross product.
The full exam-length answer is in the app.
3 marks Medium answer
Find $|\vec{a} – \vec{b}|$, if two vectors $\vec{a}$ and $\vec{b}$ are such that If $|\vec{a}| = 2$, $|\vec{b}| = 3$ and $\vec{a}. \vec{b} = 4$.
Show answer outline
$|\vec{a} - \vec{b}| = \sqrt{(|\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a}.\vec{b})}$.
The full exam-length answer is in the app.
3 marks Medium answer
Find a vector $\vec{c}$ perpendicular to both $\vec{a}$ and $\vec{b}$.
Show answer outline
Use $\vec{c} = \vec{a} \times \vec{b}$.
The full exam-length answer is in the app.
3 marks Medium answer
If \(\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k}\), \(\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}\), find \(\vec{a} + \vec{b}\) and \(\vec{a} - \vec{b}\).
Show answer outline
\(\vec{a} + \vec{b} = 4\hat{i} + 4\hat{j} + 0\hat{k}, \vec{a} - \vec{b} = 2\hat{i} + 0\hat{j} + 4\hat{k}\)
The full exam-length answer is in the app.
3 marks Medium answer
Consider the points A(1,2,7), B(2,6,3) and C(3,10, -1). Find \(\vec{AB}, \vec{BC}\).
Show answer outline
\(\vec{AB} = (1,4,-4), \vec{BC} = (1,4,-4)\)
The full exam-length answer is in the app.
3 marks Medium answer
If \(\vec{a} = \hat{i} + 3\hat{j}\) and \(\vec{b} = 3\hat{j} + \hat{k}\), then find a unit vector which is perpendicular to both \(\vec{a}\) and \(\vec{b}\).
Show answer outline
Use the cross product \(\vec{a} \times \vec{b}\).
The full exam-length answer is in the app.
3 marks Medium answer
Find the area of the parallelogram whose adjacent sides are given by the vectors \(\vec{a} = 3\hat{i} + \hat{j} + 4\hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + \hat{k}\).
Show answer outline
Area = |\vec{a} \times \vec{b}|
The full exam-length answer is in the app.
3 marks Medium answer
If \(\vec{a} = \hat{i} + 3\hat{j}\) and \(\vec{b} = 3\hat{j} + \hat{k}\), then prove that \((\vec{a}.\vec{b})^2 = \begin{vmatrix} \vec{a}.\vec{a} & \vec{a}.\vec{b} \\ \vec{a}.\vec{b} & \vec{b}.\vec{b} \end{vmatrix}$.
Show answer outline
Use properties of determinants and dot products.
The full exam-length answer is in the app.
3 marks Medium answer
Using vectors, show that the points A(1,2,7), B(2,6,3) and C(3,10,-1) are collinear.
Show answer outline
Show that \(\vec{AB} \times \vec{AC} = \vec{0}\).
The full exam-length answer is in the app.
3 marks Medium answer
Find the unit vector along \(\vec{a} - \vec{b}\), where \(\vec{a} = \hat{i} + 3\hat{j} - \hat{k}\) and \(\vec{b} = 3\hat{i} + 2\hat{j} + \hat{k}\).
Show answer outline
Unit vector = \frac{\vec{a} - \vec{b}}{|\vec{a} - \vec{b}|}
The full exam-length answer is in the app.
3 marks Medium answer
Find a unit vector in the direction of \(\vec{a} + \vec{b}\).
Show answer outline
Unit vector = \frac{\vec{a} + \vec{b}}{|\vec{a} + \vec{b}|}
The full exam-length answer is in the app.
3 marks Medium answer
Find \(\angle A\).
Show answer outline
Use the dot product to find the angle.
The full exam-length answer is in the app.
3 marks Medium answer
Find the area of triangle ABC.
Show answer outline
Area = \frac{1}{2} |\vec{AB} \times \vec{AC}|
The full exam-length answer is in the app.
3 marks Medium answer
Find the projection vector of \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\) along the vector \(\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}\$. Also write \(\vec{b}\$ as the sum of a vector along \(\vec{a}\$ and a vector perpendicular to \(\vec{a}\$.
Show answer outline
Projection: \(\frac{\vec{b} \cdot \vec{a}}{\vec{a} \cdot \vec{a}} \vec{a}\)
The full exam-length answer is in the app.
3 marks Medium answer
Find the area of a parallelogram for which the vectors \(\vec{a} = 2\hat{i} + \hat{j}\$ and \(3\hat{i} + \hat{j} + 4\hat{k}\$ are adjacent sides.
Show answer outline
Area = \(|\vec{a} \times \vec{b}|\)
The full exam-length answer is in the app.
3 marks Medium answer
If \(\vec{a}, \vec{b}\$ and \(\vec{a} + \vec{b}\$ are unit vectors, then prove that the angle between \(\vec{a}\$ and \(\vec{b}\$ is \(\frac{2\pi}{3}\$.
Show answer outline
Use \(\vec{a} \cdot \vec{b} = -\frac{1}{2}\)
The full exam-length answer is in the app.
3 marks Medium answer
If \(\vec{a} = 2\hat{i} + \hat{j} - 3\hat{k}\$ and \(m\hat{i} + 3\hat{j} - \hat{k}\$ are perpendicular to each other, then find m. Also find the area of the rectangle having these two vectors as sides.
Show answer outline
Use \(\vec{a} \cdot (m\hat{i} + 3\hat{j} - \hat{k}) = 0\)
The full exam-length answer is in the app.
3 marks Medium answer
Find the area of parallelogram with adjacent sides \(\vec{a}\$ and \(\vec{b}\$.
Show answer outline
Area = \(|\vec{a} \times \vec{b}|\)
The full exam-length answer is in the app.
3 marks Medium answer
Find the angle between the vectors \(\vec{a} = 3\hat{i} + 4\hat{j} + \hat{k}\$ and \(\vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}\$.
Show answer outline
Use \(\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\)
The full exam-length answer is in the app.
3 marks Medium answer
The adjacent sides of a parallelogram are \(\vec{a} = 3\hat{i} + \lambda\hat{j} + \hat{k}\$ and \(\vec{b} = \hat{i} + \lambda\hat{j} - \hat{k}\$. i) Find \(\vec{a} \times \vec{b}\$. ii) If the area of the parallelogram is \(\sqrt{42}\$ square units, find the value of \(\lambda\$.
Show answer outline
Area = \(|\vec{a} \times \vec{b}| = \sqrt{42}\)
The full exam-length answer is in the app.
3 marks Medium answer
The position vectors of the vertices of \(\triangle ABC\$ are \(3\hat{i} - 4\hat{j} - 4\hat{k}\$, \(2\hat{i} - \hat{j} + \hat{k}\$ and \(\hat{i} - 3\hat{j} - 5\hat{k}\$. a) Find \(\vec{AB}, \vec{BC}\$ and \(\vec{CA}\$.
Show answer outline
\(\vec{AB} = \vec{B} - \vec{A}\)
The full exam-length answer is in the app.
3 marks Medium answer
Given A(1,1,1) B(1,2,3) and C(2,3,1) are the vertices of \(\triangle ABC\$. Find the area of \(\triangle ABC\$.
Show answer outline
Area = \(\frac{1}{2}|\vec{AB} \times \vec{AC}|\)
The full exam-length answer is in the app.
3 marks Medium answer
Consider the vectors \(\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}\$ and \(\vec{b} = 6\hat{i} + 3\hat{j} + 2\hat{k}\$. a) Find \(\vec{a} \cdot \vec{b}\$ and \(\vec{a} \times \vec{b}\$.
Show answer outline
\(\vec{a} \cdot \vec{b} = 12 + 3 - 4 = 11\)
The full exam-length answer is in the app.
3 marks Medium answer
Verify that \(|\vec{a} \times \vec{b}|^2 = |\vec{a}|^2|\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2\$.
Show answer outline
Use properties of cross and dot products.
The full exam-length answer is in the app.
3 marks Medium answer
With the help of suitable figure, for any three vectors \(\vec{a}, \vec{b}, \vec{c}\$ show that \((\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})\$.
Show answer outline
Use associativity of vector addition.
The full exam-length answer is in the app.
3 marks Medium answer
If \(\vec{a} = \hat{i} - \hat{j} + \hat{k}\$, \(\vec{b} = 2\hat{i} - 2\hat{j} - \hat{k}\$, what is the projection of \(\vec{a}\$ on \(\vec{b}\$?
Show answer outline
Projection = \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b}\)
The full exam-length answer is in the app.
3 marks Medium answer
If \(\vec{a} = 3\hat{i} - \hat{j} - 5\hat{k}\$, \(\vec{b} = \hat{i} - 5\hat{j} + 3\hat{k}\$. Show that \(\vec{a} + \vec{b}\$ and \(\vec{a} - \vec{b}\$ are perpendicular.
Show answer outline
\(\vec{a} + \vec{b} \cdot \vec{a} - \vec{b} = 0\)
The full exam-length answer is in the app.
3 marks Medium answer
Let A(2,3,4), B(4,3,2) and C(5,2,-1) be three points. a) Find \(\vec{AB}, \vec{BC}\$.
Show answer outline
\(\vec{AB} = \vec{B} - \vec{A}\)
The full exam-length answer is in the app.
3 marks Medium answer
b) Find the projection of \(\vec{BC}\$ on \(\vec{AB}\$.
Show answer outline
Projection = \(\frac{\vec{BC} \cdot \vec{AB}}{|\vec{AB}|^2} \vec{AB}\)
The full exam-length answer is in the app.
3 marks Medium answer
c) Find the area of the triangle ABC.
Show answer outline
Area = \(\frac{1}{2}|\vec{AB} \times \vec{AC}|\)
The full exam-length answer is in the app.
2 marks Short answer
Write the unit vector in the direction of $\vec{a} = \hat{i} + \hat{j}$.
Show answer outline
$\frac{1}{\sqrt{2}}(\hat{i} + \hat{j})$.
The full exam-length answer is in the app.
Study it properly — free, in the app
The full Veda Bites deck, complete notes, spaced-repetition flashcards, leveled MCQs, tests and games for this kit — plus Daily Facts and the Arena, every day.