Sample QuestionsAlgebra of Vectors questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The position vectors of the points $A, B, C$ are $2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\ 3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+4\hat{\text{j}}-3\hat{\text{k}}$ respectively. These points,
- ✓
Form an isosceles triangle.
- B
- C
- D
Answer: A.
View full solution →If $\text{G}$ is the intersection of diagonals of a parallelogram $\text{ABCD}$ and $\text{O}$ is any point, then $\overrightarrow{\text{OA}}+\overrightarrow{\text{OB}}+\overrightarrow{\text{OC}}+\overrightarrow{\text{OD}}=$
- A
$2\overrightarrow{\text{OG}}$
- ✓
$4\overrightarrow{\text{OG}}$
- C
$5\overrightarrow{\text{OG}}$
- D
$3\overrightarrow{\text{OG}}$
Answer: B.
View full solution →$\text{ABCD}$ is a parallelogram with $\text{AC}$ and $\text{BD}$ as diagonals. Then, $\overrightarrow{\text{AC}}-\overrightarrow{\text{BD}}=$
- A
$4\overrightarrow{\text{AB}}$
- B
$3\overrightarrow{\text{AB}}$
- ✓
$2\overrightarrow{\text{AB}}$
- D
$\overrightarrow{\text{AB}}$
Answer: C.
View full solution →In a regular hexagon $\text{ABCDEF,} \overrightarrow{\text{AB}}=\vec{\text{a}},\ \overrightarrow{\text{BC}}=\vec{\text{b}}$ and $\overrightarrow{\text{CD}}=\vec{\text{c}}$. Then, $\overrightarrow{\text{AE}}=$
- A
$\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}$
- B
$2\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}$
- ✓
$\vec{\text{b}}+\vec{\text{c}}$
- D
$\vec{\text{a}}+2\vec{\text{b}}+2\vec{\text{c}}$
Answer: C.
View full solution →If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two collinear vectors, then which of the follwoing are incorrect?
- A
$\vec{\text{b}}=\lambda\vec{\text{a}}$ for some scalar $\lambda$
- B
$\vec{\text{a}}=\pm\vec{\text{b}}$
- C
The respective components of $\vec{\text{a}}$ and $\vec{\text{b}}$ are proportional.
- ✓
Both the vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ have the same direction but different magnitudes.
Answer: D.
View full solution →Represent the following graphically:
- A displacement of 40km, 30º east of north.
- A displacement of 50km south-east.
- A displacement of 70km, 40º north of west.
Find the angle at which the following vectors are inclined to each of the coordinate axes:
$4\hat{\text{i}}+8\hat{\text{j}}+\hat{\text{k}}$
View full solution →Find the sum of the following vectors: $\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}},\vec{\text{b}}=2\hat{\text{i}}-3\hat{\text{j}},\vec{\text{c}}=2\hat{\text{i}}-3\hat{\text{k}}$.
View full solution →Find the position vector of the mid-point of the vector joining the points P(2, 3, 4) and Q(4, 1, -2).
For what value of 'a' the vectors $2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ and $\text{a}\hat{\text{i}}+6\hat{\text{j}}-8\hat{\text{k}}$ are collinear?
View full solution →Find the coordinates of the tip of the position vector which is equivalent to $\overrightarrow{\text{AB}}$, where the coordinates of A and B are (-1, 3) and (-2, 1) respectively.
View full solution →Prove that the sum of all vectors drawn from the centre of a regular octagon to its vertices is the zero vector.
Find the position vector of a point R which divides the line joining the two points P and Q with position vectors $\overrightarrow{\text{OP}}=2\vec{\text{a}}+\vec{\text{b}}\text{ and }\overrightarrow{\text{OQ}}=\vec{\text{a}}-2\vec{\text{b}}$, respectively in the ratio 1 : 2 internally and externally.
View full solution →Prove that the given vectors are coplanar:
$\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ 2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}$ and $-\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}$
View full solution →ABCDE is a pentagon, prove that,$\overrightarrow{\text{AB}}+\overrightarrow{\text{AE}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{ED}}+\overrightarrow{\text{AC}}=3\ \overrightarrow{\text{AC}}$
View full solution →Prove that the points $\hat{\text{i}}-\hat{\text{j}},\ 4\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}$ and $2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ are the vertices of a right-angled triangle.
View full solution →The adjacent sides of a parallelogram are represented by the vectors $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{b}}=-2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$. Find the unit vectors parallel to the diagonals of the parallelogram.
View full solution →If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are non-zero, non-coplanar vectors, prove that the vector is coplanar:
$5\vec{\text{a}}+6\vec{\text{b}}+7\vec{\text{c}},\ 7\vec{\text{a}}-8\vec{\text{b}}+9\vec{\text{c}}$ and $3\vec{\text{a}}+20\vec{\text{b}}+5\vec{\text{c}}$
View full solution →If $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are non-coplanar vectors, prove that the point having the following position vectors is collinear:$\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}},\ 4\vec{\text{a}}+3\vec{\text{b}},\ 10\vec{\text{a}}+7\vec{\text{b}}-2\vec{\text{c}}$
View full solution →If a unit vector $\vec{\text{a}}$ makes an angle $\frac{\pi}3$ with $\hat{\text{i}}$, $\frac{\pi}4$ with $\hat{\text{j}}$ and an acute angle $\theta$ with $\hat{\text{k}}$, then find $\theta$ and hence, the components of $\vec{\text{a}}$.
View full solution →