Vector Basics PYQs
Practice Class 12 CBSE Board Previous Year Questions (2008-2026)
Q1
2008 Board
00:00
Find the magnitude of the vector $\vec{a} = 2\hat{i} - 3\hat{j} + 6\hat{k}$. 1 Mark
Step 1: Identify Components
The vector is $\vec{a} = 2\hat{i} - 3\hat{j} + 6\hat{k}$. Here $a_x = 2, a_y = -3, a_z = 6$.
Step 2: Apply Magnitude Formula
$|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}$.
Step 3: Calculate
$|\vec{a}| = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7$.
7
Q2
2008 Board
00:00
Find a unit vector in the direction of $\vec{a} = 3\hat{i} + 4\hat{j}$. 1 Mark
Step 1: Find Magnitude
$|\vec{a}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
Step 2: Apply Unit Vector Formula
Unit vector $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$.
Step 3: Calculate
$\hat{a} = \frac{3\hat{i} + 4\hat{j}}{5} = \frac{3}{5}\hat{i} + \frac{4}{5}\hat{j}$.
(3/5)i + (4/5)j
Q3
2012 Board
00:00
Find the magnitude of $(3\hat{i} + 4\hat{j}) + (\hat{i} - 2\hat{j})$. 1 Mark
Step 1: Perform Addition
Let $\vec{r} = (3\hat{i} + 4\hat{j}) + (\hat{i} - 2\hat{j}) = 4\hat{i} + 2\hat{j}$.
Step 2: Calculate Magnitude
$|\vec{r}| = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$.
2√5
Q4
2014 Board
00:00
Find the direction cosines of the vector $2\hat{i} - 3\hat{j} + 6\hat{k}$. 1 Mark
Step 1: Find Magnitude
$|\vec{r}| = \sqrt{2^2 + (-3)^2 + 6^2} = 7$.
Step 2: Direction Cosines Formula
$l = x/r, m = y/r, n = z/r$.
Step 3: Calculate
$l = 2/7, m = -3/7, n = 6/7$.
(2/7, -3/7, 6/7)
Q5
2019 Board
00:00
Find a unit vector in the direction of $2\hat{i} - \hat{j} + 2\hat{k}$. 1 Mark
Step 1: Magnitude
Magnitude $= \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{9} = 3$.
Step 2: Unit Vector
$\hat{n} = \frac{1}{3}(2\hat{i} - \hat{j} + 2\hat{k})$.
(1/3)(2i - j + 2k)
Q6
2012 Board
00:00
Find the vector joining the points $A(1,2,3)$ and $B(4,5,6)$. 1 Mark
Step 1: Position Vectors
$\vec{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{OB} = 4\hat{i} + 5\hat{j} + 6\hat{k}$.
Step 2: Calculate AB
$\vec{AB} = \vec{OB} - \vec{OA} = (4-1)\hat{i} + (5-2)\hat{j} + (6-3)\hat{k}$.
$\vec{AB} = 3\hat{i} + 3\hat{j} + 3\hat{k}$.
3i + 3j + 3k
Q7
2026 Board
00:00
Find the vector of magnitude 14 in the direction of \vec{QP}, where P and Q are the points (1, 3, 2) and (−1, 0, 8) respectively. 2 Marks
Solution will be updated soon.
Q8
2025 Board
00:00
Find $\vec{a}$ vector of magnitude 21 units in the direction opposite to that of \vec{AB} where A and B are the points A(2, 1, 3) and B(8, –1, 0) respectively. 2 Marks
Solution will be updated soon.
Q9
2023 Board
00:00
The magnitude of the vector $6\hat{i}$ − $2\hat{j}$ + $3\hat{k}$ is: 1 Mark
Solution will be updated soon.
Q10
2016 Board
00:00
If $\vec{a}$ = $4\hat{i}$ – $\hat{j}$ + $\hat{k}$ and $\vec{b}$ = $2\hat{i}$ – $2\hat{j}$ + $\hat{k}$, then find $\vec{a}$ unit vector parallel to the vector $\vec{a}$ + b. 1 Mark
Solution will be updated soon.
Q11
2026 Board
00:00
The value of m for which the points with position vectors −$\hat{i}$ − $\hat{j}$ + $2\hat{k}$, $2\hat{i}$ + $m\hat{j}$ + $5\hat{k}$ and $3\hat{i}$ + $11\hat{j}$ + $6\hat{k}$ are collinear, is ______. 1 Mark
Solution will be updated soon.
Q12
2024 Board
00:00
The position vectors of points P and Q are $\vec{p}$ and $\vec{q}$ respectively. The point R divides line segment PQ in the ratio 3 : 1 and S is the $mid-point$ of line segment PR. The position vector of S is ______. 1 Mark
Solution will be updated soon.
Q13
2023 Board
00:00
Two vectors $\vec{a}$ = a₁$\hat{i}$ + a₂$\hat{j}$ + a₃$\hat{k}$ and $\vec{b}$ = b₁$\hat{i}$ + b₂$\hat{j}$ + b₃$\hat{k}$ are collinear if: 1 Mark
Solution will be updated soon.
Q14
2019 Board
00:00
Show that the points A(−$2\hat{i}+5\hat{j}$+$5\hat{k}$), B($\hat{i}+2\hat{j}$+$5\hat{k}$) and C($7\hat{i}$−$\hat{k}$) are collinear. 1 Mark
Solution will be updated soon.
Q15
2016 Board
00:00
Find the position vector of $\vec{a}$ point which divides the join of points with position vectors a−2b and $2\vec{a}+\vec{b}$ externally in the ratio 2:1. 1 Mark
Solution will be updated soon.
Q16
2026 Board
00:00
Vectors $\vec{a}$ = $3\hat{i}$ − $2\hat{j}$ + $2\hat{k}$ and $\vec{b}$ = $\hat{i}$ + $2\hat{k}$ represent the two adjacent sides of $\vec{a}$ parallelogram. Find the vectors representing its diagonals and hence find their lengths. 2 Marks
Solution will be updated soon.
Q17
2024 Board
00:00
Assertion–Reason: Assertion (A) gives vectors $\vec{a}$ = $6\hat{i}+2\hat{j}$−$8\hat{k}$, $\vec{b}$ = $10\hat{i}$−$2\hat{j}$−$6\hat{k}$, $\vec{c}$ = $4\hat{i}$−$4\hat{j}+2\hat{k}$ (these form $\vec{a}$ triangle); Reason (R): "Three $non-zero$ vectors of which none of two are collinear forms $\vec{a}$ triangle if their resultant is zero vector or sum of any two vectors is equal to the third. 1 Mark
Solution will be updated soon.
Q18
2016 Board
00:00
The two adjacent sides of $\vec{a}$ parallelogram are $2\hat{i}$–$4\hat{j}$–$5\hat{k}$ and $2\hat{i}+2\hat{j}$+$3\hat{k}$. Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram. 1 Mark
Solution will be updated soon.
Q19
2016 Board
00:00
The two vectors $\hat{j}+\hat{k}$ and $3\hat{i}$–$\hat{j}+4\hat{k}$ represent the two sides AB and AC respectively of triangle ABC. Find the length of the median through A. 1 Mark
Solution will be updated soon.