Vector Algebra

Comprehensive Question Bank & Study Notes

Multiple Choice Questions

1 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#570
If vector \(\vec{a} = 3\hat{i} + 2\hat{j} - \hat{k}\) and vector \(\vec{b} = \hat{i} - \hat{j} + \hat{k}\), then which of the following is correct ?
(A) \(\vec{a} \parallel \vec{b}\)
(B) \(\vec{a} \perp \vec{b}\)
(C) \(|\vec{b}| > |\vec{a}|\)
(D) \(|\vec{a}| = |\vec{b}|\)
MEDIUM
2 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#569
If \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), \(|\vec{a}| = \sqrt{37}\), \(|\vec{b}| = 3\) and \(|\vec{c}| = 4\), then the angle between \(\vec{b}\) and \(\vec{c}\) is
(A) \(\dfrac{\pi}{6}\)
(B) \(\dfrac{\pi}{4}\)
(C) \(\dfrac{\pi}{3}\)
(D) \(\dfrac{\pi}{2}\)
MEDIUM
3 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#568
The projection vector of vector \(\vec{a}\) on vector \(\vec{b}\) is
(A) \((\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^{2}})\vec{b}\)
(B) \(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\)
(C) \(\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|}\)
(D) \((\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|^{2}})\vec{b}\)
MEDIUM
4 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#567
Let \(\vec{p}\) and \(\vec{q}\) be two unit vectors and \(\alpha\) be the angle between them. Then \((\vec{p}+\vec{q})\) will be a unit vector for what value of \(\alpha\)?
(A) \(\frac{\pi}{4}\)
(B) \(\frac{\pi}{3}\)
(C) \(\frac{\pi}{2}\)
(D) \(\frac{2\pi}{3}\)
MEDIUM
5 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#566
If the sides AB and AC of \(\triangle ABC\) are represented by vectors \(\hat{j}+\hat{k}\) and \(3\hat{i}-\hat{j}+4\hat{k}\) respectively, then the length of the median through A on BC is:
(A) \(2\sqrt{2}\) units
(B) \(\sqrt{18}\) units
(C) \(\frac{\sqrt{34}}{2}\) units
(D) \(\frac{\sqrt{48}}{2}\) units
MEDIUM
6 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#565
Let \(\vec{a}\) be a position vector whose tip is the point \((2,-3)\). If \(\vec{AB}=\vec{a}\), where coordinates of A are \((-4, 5)\), then the coordinates of B are:
(A) \((-2,-2)\)
(B) \((2,-2)\)
(C) \((-2,2)\)
(D) \((2, 2)\)
MEDIUM
7 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#564
The respective values of \(|\vec{a}|\) and \(|\vec{b}|\), if given \((\vec{a}-\vec{b})\cdot(\vec{a}+\vec{b})=512\) and \(|\vec{a}|=3|\vec{b}|\), are:
(A) 48 and 16
(B) 3 and 1
(C) 24 and 8
(D) 6 and 2
MEDIUM
8 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#563
Let \(|\vec{a}|=5\) and \(-2\le\lambda\le1\). Then, the range of \(|\lambda\vec{a}|\) is:
(A) [5, 10]
(B) [-2, 5]
(C) [2, 1]
(D) [-10, 5]
MEDIUM
9 Competency APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#562
A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector \(3\hat{i}+15\hat{j}+6\hat{k}\) and the other is along the vector \(2\hat{i}+10\hat{j}+\lambda\hat{k}\), then the value of \(\lambda\) is :
(A) 6
(B) 1
(C) \(\frac{1}{4}\)
(D) 4
MEDIUM
10 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#561
If \(|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|\) for any two vectors, then vectors \(\vec{a}\) and \(\vec{b}\) are:
(A) orthogonal vectors
(B) parallel to each other
(C) unit vectors
(D) collinear vectors
MEDIUM
11 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#560
Let \(\theta\) be the angle between two unit vectors \(\hat{a}\) and \(\hat{b}\) such that \(\sin\theta=\frac{3}{5}\). Then, \(\hat{a} \cdot \hat{b}\) is equal to:
(A) \(\pm\frac{3}{5}\)
(B) \(\pm\frac{3}{4}\)
(C) \(\pm\frac{4}{5}\)
(D) \(\pm\frac{4}{3}\)
MEDIUM
12 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#559
The vector with terminal point \(A(2,-3,5)\) and initial point \(B(3, 4, 7)\) is:
(A) \(\hat{i}-\hat{j}+2\hat{k}\)
(B) \(\hat{i}+\hat{j}+2\hat{k}\)
(C) \(-\hat{i}-\hat{j}-2\hat{k}\)
(D) \(-\hat{i}+\hat{j}-2\hat{k}\)
EASY
13 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#558
For any two vectors \(\vec{a}\) and \(\vec{b}\), which of the following statements is always true?
(A) \(\vec{a}.\vec{b}\ge
(B) \vec{a}
(C)
(D) \vec{b}
MEDIUM
14 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#557
The unit vector perpendicular to both vectors \(\hat{i}+\hat{k}\) and \(\hat{i}-\hat{k}\) is:
(A) \(2\hat{j}\)
(B) \(\hat{j}\)
(C) \(\frac{\hat{i}-\hat{k}}{\sqrt{2}}\)
(D) \(\frac{\hat{i}+\hat{k}}{\sqrt{2}}\)
MEDIUM
15 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#556
If \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}-\hat{k}\), then \(\vec{a}\) and \(\vec{b}\):
(A) collinear vectors which are not parallel
(B) parallel vectors
(C) perpendicular vectors
(D) unit vectors
MEDIUM
16 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#555
If \(|\vec{a}|= 2\) and \(-3\le k\le2\), then \(|\vec{k}\vec{a}|\in\):
(A) [-6, 4]
(B) [0, 4]
(C) [4, 6]
(D) [0, 6]
MEDIUM
17 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#554
If \(\vec{a}\) and \(\vec{b}\) are two vectors such that \(|\vec{a}|=1,|\vec{b}|=2~and\vec{a}\cdot\vec{b}=\sqrt{3}\) then the angle between \(2\vec{a}\) and \(-\vec{b}\) is:
(A) \(\frac{\pi}{6}\)
(B) \(\frac{\pi}{3}\)
(C) \(\frac{5\pi}{6}\)
(D) \(\frac{11\pi}{6}\)
MEDIUM
18 Competency APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#553
The vectors \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\), \(\vec{b}=\hat{i}-3\hat{j}-5\hat{k}\) and \(\vec{c}=-3\hat{i}+4\hat{j}+4\hat{k}\) represents the sides of
(A) an equilateral triangle
(B) an obtuse-angled triangle
(C) an isosceles triangle
(D) a right-angled triangle
MEDIUM
19 Competency APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#552
Let \(\vec{a}\) be any vector such that \(|\vec{a}|=a\) The value of \(|\vec{a}\times\hat{i}|^{2}+|\vec{a}\times\hat{j}|^{2}+|\vec{a}\times\hat{k}|^{2}\) is:
(A) \(a^{2}\)
(B) \(2a^{2}\)
(C) \(3a^{2}\)
(D) 0
MEDIUM
20 Competency APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#551
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:
(A) \(\frac{\vec{p}+3\vec{q}}{4}\)
(B) \(\frac{\vec{p}+3\vec{q}}{8}\)
(C) \(\frac{5\vec{p}+3\vec{q}}{4}\)
(D) \(\frac{5\vec{p}+3\vec{q}}{8}\)
MEDIUM
21 Competency APPLY MCQ_SINGLE 1 Mark(s) 2023
#860
In $\Delta ABC$, $\vec{AB}=\hat{i}+\hat{j}+2\hat{k}$ and $\vec{AC}=3\hat{i}-\hat{j}+4\hat{k}$. If D is mid-point of BC, then vector $\vec{AD}$ is equal to :
(A) $4\hat{i}+6\hat{k}$
(B) $2\hat{i}-2\hat{j}+2\hat{k}$
(C) $\hat{i}-\hat{j}+\hat{k}$
(D) $2\hat{i}+3\hat{k}$
MEDIUM
22 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#859
$\vec{a}$ and $\vec{b}$ are two non-zero vectors such that the projection of $\vec{a}$ on $\vec{b}$ is 0. The angle between $\vec{a}$ and $\vec{b}$:
(A) $\frac{\pi}{2}$
(B) $\pi$
(C) $\frac{\pi}{4}$
(D) 0
MEDIUM
23 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#858
If a vector makes an angle of $\frac{\pi}{4}$ with the positive directions of both x-axis and y-axis, then the angle which it makes with positive z-axis is :
(A) $\frac{\pi}{4}$
(B) $\frac{3\pi}{4}$
(C) $\frac{\pi}{2}$
(D) 0
MEDIUM
24 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2023
#857
13. If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$ then $\vec{a} \cdot \vec{b} \ge 0$ only when:
(A) $0 < \theta < \frac{\pi}{2}$
(B) $0 \le \theta \le \frac{\pi}{2}$
(C) $0 < \theta < \pi$
(D) $0 \le \theta \le \pi$
MEDIUM
25 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#856
A unit vector along the vector $4\hat{i}-3\hat{k}$ is:
(A) $\frac{1}{7}(4\hat{i}-3\hat{k})$
(B) $\frac{1}{5}(4\hat{i}-3\hat{k})$
(C) $\frac{1}{\sqrt{7}}(4\hat{i}-3\hat{k})$
(D) $\frac{1}{\sqrt{5}}(4\hat{i}-3\hat{k})$
EASY
26 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023 AISSCE(Board Exam)
#820
If $\vec{a}+\vec{b}=\hat{i}$ and $\vec{a}=2\hat{i}-2\hat{j}+2\hat{k}$, then |$\vec{b}$| equals:
(A) \sqrt{14}
(B) 3
(C) \sqrt{12}
(D) \sqrt{17}
MEDIUM
27 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2023 AISSCE(Board Exam)
#819
The value of $(\hat{i}\times\hat{j})\cdot \hat{j}+(\hat{j}\times\hat{i}) \hat{k}:$
(A) 2
(B) 0
(C) 1
(D) -1
EASY
28 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023 AISSCE(Board Exam)
#818
The value of p for which the vectors $2\hat{i}+p\hat{j}+\hat{k}$ and $-4\hat{i}-6\hat{j}+26\hat{k}$ are perpendicular to each other, is:
(A) 3
(B) -3
(C) $-\frac{17}{3}$
(D) $\frac{17}{3}$
MEDIUM
29 Knowledge REMEMBER MCQ_SINGLE 1 Mark(s) 2023
#817
The magnitude of the vector 6î - 2î + 3ê is
(A) 1
(B) 5
(C) 7
(D) 12
EASY
30 Knowledge REMEMBER MCQ_SINGLE 1 Mark(s) 2023 AISSCE(Board Exam)
#816
Two vectors $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ are collinear if
(A) a₁b₁ + a₂b₂ + a₃b₃ = 0
(B) a₁/b₁ = a₂/b₂ = a₃/b₃
(C) a₁=b₁, a₂=b₂, a₃=b₃
(D) a₁+a₂+a₃ = b₁+b₂+b₃
EASY

Assertion & Reason

1 Competency APPLY ASSERTION_REASON 1 Mark(s) 2025 AISSCE(Board Exam)
#976
Assertion: Assertion (A) : If $|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}=256$ and $|\vec{b}|=8$, then $|\vec{a}|=2$.
Reason: Reason (R) : $\sin ^{2} \theta+\cos ^{2} \theta=1$ and $|\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \sin \theta$ and $\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta$.
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is NOT the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.
MEDIUM

Very Short Answer (2 Mark)

1 Knowledge REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1448
Vector $\vec{r}$ is inclined at equal angles to the three axes x, y and z. If magnitude of $\vec{r}$ is $5\sqrt{3}$ units, then find $\vec{r}$.
MEDIUM
2 Knowledge REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1447
If $\vec{a}$ and $\vec{b}$ are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that $BC=3BA$.
MEDIUM
3 Knowledge UNDERSTAND VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1426
Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three vectors such that $\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}$ and $\vec{a}\times\vec{b}=\vec{a}\times\vec{c}$, $\vec{a}\ne\vec{0}$. Show that $\vec{b}=\vec{c}$.
MEDIUM
4 Knowledge REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1425
Find a vector of magnitude 5 which is perpendicular to both the vectors $3\hat{i}-2\hat{j}+\hat{k}$ and $4\hat{i}+3\hat{j}-2\hat{k}$.
MEDIUM
5 Knowledge REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1381
If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors, then find x, such that $\vec{\alpha}=(x-2)\vec{a}+\vec{b}$ and $\vec{\beta}=(3+2x)\vec{a}-2\vec{b}$ are collinear.
MEDIUM
6 Knowledge REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1378
If $\vec{\alpha}$ and $\vec{\beta}$ are position vectors of two points P and Q respectively, then find the position vector of a point R in QP produced such that $QR=\frac{3}{2}QP$.
MEDIUM
7 Knowledge REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1377
A vector $\vec{a}$ makes equal angles with all the three axes. If the magnitude of the vector is $5\sqrt{3}$ units, then find $\vec{a}$.
MEDIUM
8 Knowledge REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1361
Find 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.
MEDIUM
9 Competency REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1360
Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors $\vec{a}=3\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+4\hat{k}$. Determine the angle formed between the kite strings. Assume there is no slack in the strings.
EASY
10 Knowledge REMEMBER VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1358
The diagonals of a parallelogram are given by $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$. Find the area of the parallelogram.
MEDIUM
11 Competency UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1294
In the given figure, ABCD is a parallelogram. If $\vec{AB}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{DB}=3\hat{i}-6\hat{j}+2\hat{k}$ , then find $\vec{AD}$ and hence find the area of parallelogram ABCD.
MEDIUM
12 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1293
If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $(\vec{a}+\vec{b})\perp\vec{a}$ and $(2\vec{a}+\vec{b})\perp\vec{b}$ , then prove that $|\vec{b}|=\sqrt{2}|\vec{a}|$.
MEDIUM
13 Knowledge REMEMBER VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1250
Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors. Prove that $|\vec{a}\times\vec{b}|\le|\vec{a}||\vec{b}|$ . State the condition under which equality holds, i.e., $|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|$
EASY
14 Knowledge REMEMBER VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1249
Find the position vector of point C which divides the line segment joining points A and B having position vectors $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively in the ratio $4:1$ externally. Further, find $|\vec{AB}|:|\vec{BC}|$ .
MEDIUM
15 Knowledge APPLY VSA 2 Mark(s) 2023
#926
If $\vec{r}=3\hat{i}-2\hat{j}+6\hat{k}$, find the value of $(\vec{r}\times\hat{j})\cdot(\vec{r}\times\hat{k})-12$
MEDIUM
16 Knowledge APPLY VSA 2 Mark(s) 2023
#925
24. If the projection of the vector $\hat{i}+\hat{j}+\hat{k}$ on the vector $p\hat{i}+\hat{j}-2\hat{k}$ is $\frac{1}{3}$, then find the value(s) of $p$.
MEDIUM
17 Knowledge APPLY VSA 2 Mark(s) 2023
#876
Find all the vectors of magnitude $3\sqrt{3}$ which are collinear to vector $\hat{i}+\hat{j}+\hat{k}.$
MEDIUM
18 Knowledge APPLY VSA 2 Mark(s) 2023 AISSCE(Board Exam)
#875
(a) If the vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a}| = 3$, $|\vec{b}| = \frac{2}{3}$ and $\vec{a} \times \vec{b}$ is a unit vector, then find the angle between $\vec{a}$ and $\vec{b}$.
OR(b) Find the area of a parallelogram whose adjacent sides are determined by the vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$.
MEDIUM

Short Answer (3 Marks)

1 Knowledge REMEMBER SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1479
The scalar product of the vector $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$ with a unit vector along sum of vectors $\vec{b}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{c}=\lambda\hat{i}-2\hat{j}-3\hat{k}$ is equal to 1. Find the value of $\lambda$.
MEDIUM
2 Knowledge REMEMBER SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1434
If $\vec{a}$ and $\vec{b}$ are unit vectors inclined with each other at an angle $\theta$, then prove that $\frac{1}{2}|\vec{a}-\vec{b}|=\sin\frac{\theta}{2}$.
MEDIUM
3 Knowledge REMEMBER SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1433
If $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ such that $|\vec{a}|=3, |\vec{b}|=5, |\vec{c}|=7$, then find the angle between $\vec{a}$ and $\vec{b}$.
MEDIUM
4 Competency UNDERSTAND SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1367
During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by $\vec{B}=2\hat{i}+8\hat{j}$, $\vec{W}=6\hat{i}+12\hat{j}$ and $\vec{F}=12\hat{i}+18\hat{j}$ respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.
MEDIUM
5 Competency UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1347
Find a vector of magnitude 4 units perpendicular to each of the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$ and hence verify your answer.
MEDIUM
6 Competency UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1279
The position vectors of vertices of $\Delta$ ABC are $A(2\hat{i}-\hat{j}+\hat{k}),$ $B(\hat{i}-3\hat{j}-5\hat{k})$ and $C(3\hat{i}-4\hat{j}-4\hat{k})$ Find all the angles of $\Delta$ AВС.
HARD

Long Answer (5 Marks)

1 Knowledge REMEMBER LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1419
Show that the area of a parallelogram whose diagonals are represented by $\vec{a}$ and $\vec{b}$ is given by $\frac{1}{2}|\vec{a}\times\vec{b}|$. Also find the area of a parallelogram whose diagonals are $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}-\hat{k}$.
MEDIUM

Case Study & Passage Based

Read the following passage:
Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his predecided destination, following straight paths from A to C and B to C in such a way that $\overrightarrow{OA} = \vec{a}$, $\overrightarrow{OB} = \vec{b}$ and $\overrightarrow{OC} = 5\vec{a}-2\vec{b}$ respectively.
1 Knowledge SUBJECTIVE 4 Mark(s) 2025 AISSCE(Board Exam)
#965
(i) Complete the given figure to explain their entire movement plan along the respective vectors.
(ii) Find vectors $\vec{AC}$ and $\vec{BC}$.
(iii) (a) If $\vec{a} \cdot \vec{b} = 1$, distance of O to A is 1 km and that from O to B is 2 km, then find the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$. Also, find $|
\vec{a} \times \vec{b}|$.
OR
(iii) (b) If $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$, then find a unit vector perpendicular to $(\vec{a}+\vec{b})$ and $(\vec{a}-\vec{b})$.