Sample QuestionsVector Algebra 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 points $P$ and $Q$ are $\vec{p}$ and $\vec{q}$ respectively. The point $R$ divides line segment $P Q$ in the ratio $3: 1$ and $S$ is the mid-point of line segment $P R$. 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
$\frac{5 \vec{p}+3 \vec{q}}{8}$
View full solution →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
$\frac{11 \pi}{6}$
View full solution →The unit vector perpendicular to both vectors $\hat{i}+\hat{k}$ and $\hat{i}-\hat{k}$ is:
View full solution →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
- B
an obtuse-angled triangle
- C
- D
View full solution →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
$2 a^2$
- C
$3 a^2$
- D
$0$
View full solution →Assertion $(A)$ : The vectors :
\[\vec{a}=6 \hat{i}+2 \hat{j}-8 \hat{k}, \vec{b}=10 \hat{i}-2 \hat{j}-6 \hat{k}, \vec{c}=4 \hat{i}-4 \hat{j}+2 \hat{k}\]
represent the sides of a right angled triangle.
Reason $(R)$ : Three non-zero vectors of which none of two are collinear forms a triangle if their resultant is zero vector or sum of any two vectors is equal to the third.
- A
Both Assertion (A) and Reason (R) are true and the Reason $(R)$ is the correct explanation of the Assertion (A).
- B
Both Assertion (A) and Reason $(R)$ are true but Reason $(R)$ is not the correct explanation of the Assertion (A).
- C
Assertion (A) is true but Reason (R) is false.
- D
Assertion (A) is false but Reason $( R )$ is true.
View full solution →Assertion (A): For two hon-zero vectors $\vec{a}$ and $b , \vec{a} \cdot b = b \cdot \vec{a}$.
Reason (R): For two non-zero vectors $\vec{a}$ and $\vec{b}, \vec{a} \times \vec{b}=\vec{b} \times \vec{a}$.
- A
Both Assertion (A) and Reason (R) are true and Reason $(R)$ is the correct explanation of the Assertion (A).
- B
Both Assertion (A) and Reason (R) are true, but Reason $(R)$ is not the correct explanation of the Assertion (A).
- C
Assertion (A) is true but Reason (R) is false.
- D
Assertion (A) is false but Reason (R) is true.
View full solution →Assertion (A): $(\vec{b} \cdot \vec{c}) \vec{a}$ is a scalar quantity.
Reason $(R)$ : Dot product of two vectors is a scalar quantity.
- A
Both Assertion (A) and Reason (R) are true and Reason $(R)$ is the correct explanation of the Assertion (A).
- B
Both Assertion (A) and Reason (R) are true but Reason $(R)$ is not the correct explanation of the Assertion (A).
- C
Assertion (A) is true but Reason (R) is false.
- D
Assertion $( A )$ is false but Reason $( R )$ is true.
View full solution →Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The magnitude of the resultant of vectors $\overline{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$
Reason: The magnitude of a vector can never be negative.
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
View full solution →Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The adjacent sides of a parallelogramarealong $\overline{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}$ and $\overline{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}$ The angle between the diagonals is $150^\circ$.
Reason: Two vectors are perpendicular to each other if their dot product is zero.
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
View full solution →If $\theta $ is the angle between any two vectors $\vec a$ and $\vec b$, then $\left| {\vec a.\vec b} \right| = \left| {\vec a \times \vec b} \right|$ when θ is equal to
View full solution →The value of $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$ is
View full solution →Let $\vec a$ and $\vec b$ be two unit vectors and θ is the angle between them. Then $\vec a + \vec b$ is a unit vector if
View full solution →If $\theta $ is the angle between two vectors$\;\vec a\;$and $\vec b$, then $\vec a.\vec b \geq 0$ only when
View full solution →Area of a rectangle having vertices A, B, C and D with position vectors $ - \hat i + \frac{1}{2}\hat j + 4\hat k$, $\hat i + \frac{1}{2}\hat j + 4\hat k$, $\hat i - \frac{1}{2}\hat j + 4\hat k$ and $- \hat i - \frac{1}{2}\hat j + 4\hat k$, respectively is
View full solution →If $\vec a = \hat i + \hat j + \hat k,\vec b = 2\hat i - \hat j + 3\hat k$ and $\vec c = \hat i - 2\hat j + \hat k$ find a unit vector parallel to the vector $2\vec a - \vec b + 3\vec c$
View full solution →Find the value of x for which $x\left( {\hat i + \hat j + \hat k} \right)$ is a unit vector.
View full solution →If $\vec a = \vec b + \vec c$, then is it true that $\left| {\vec a} \right| = \left| {\vec b} \right| + \left| {\vec c} \right|$? Justify your answer.
View full solution →Find the scalar components and magnitude of the vector joining the points P(x1, y1, z1) and Q (x2, y2, z2)
Prove that $(\vec{a}+\vec{b}) \cdot(\vec{a}+\vec{b})=|\vec{a}|^{2}+|\vec{b}|^{2}$, if and only if $\vec{a}, \vec{b}$ are perpendicular, given $\vec{a} \neq \vec{0}, \vec{b} \neq \vec{0}$.
View full solution →Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $\left( {2\vec a + \vec b} \right)$and$\left( {\vec a - 3\vec b} \right)$ externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.
View full solution →Find a vector of magnitude 5 units and parallel to the resultant of the vectors $\vec a = 2\hat i + 3\hat j - \hat k$ and $\vec b = \hat i - 2\hat j + \hat k$
View full solution →A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.
Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a}=\hat{i}-\hat{j}+3 \hat{k}$ and $\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}$.
View full solution →Show that the points A (1, -2, -8), B (5, 0, -2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
If $\vec{a}, \vec{b}, \vec{\mathrm{c}}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec{c} \cdot \vec{d}=15$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec c$.
View full solution →The scalar product of the vector $\hat i + \hat j + \hat k$ with a unit vector along the sum of vectors $2\hat i + 4\hat j - 5\hat k\;$ and $\lambda \hat i + 2\hat j + 3\hat k$ is equal to one. Find the value of$\;\lambda $.
View full solution →Let $\vec a = \hat i + 4\hat j + 2\hat k$, $\vec b = 3\hat i - 2\hat j + 7\hat k$ and $\vec c = 2\hat i - \hat j + 4\hat k$. Find a vector $\vec d$ which is perpendicular to both $\vec a$ and $\vec b$, and $\vec c.\vec d = 15$.
View full solution →The two adjacent sides of a parallelogram are $2\hat i - 4\hat j + 5\hat k\;$ and $\hat i - 2\hat j - 3\hat k$. Find the unit vector parallel to its diagonal. Also, find its area.
View full solution →Read the following passage and answer the questions given below: Teams A,B,C went for playing a tug of war game. Teams A,B,C have attached a rope to a metal ring and is trying to pull the ring into their own area.
TeamApulls with force $F_1=6 \hat{i}+0 \hat{j} k N$
TeamBpulls with force $F_2=-4 \hat{i}+4 \hat{j} k N$
TeamCpulls with force $F_3=-3 \hat{i}-3 \hat{j} k N$,
(i) What is the magnitude of the force of Team A?
(ii) Which team will win the game?
(iii) Find the magnitude of the resultant force exerted by the teams.
OR
(iii) In what direction is the ring getting pulled?
View full solution →Ishaan left from his village on weekend. First, he travelled up to temple. After this, he left for the zoo. After this, he left for shopping in a mall. The positions of Jshaan at different places is given in the following graph.
Based on the above information, answer the following questions.
- Position vector of B is:
- $3\hat{\text{i}}+5\hat{\text{j}}$
- $5\hat{\text{i}}+3\hat{\text{j}}$
- $-5\hat{\text{i}}-3\hat{\text{j}}$
- $-5\hat{\text{i}}+3\hat{\text{j}}$
- Position vector of D is:
- $5\hat{\text{i}}+3\hat{\text{j}}$
- $3\hat{\text{i}}+5\hat{\text{j}}$
- $8\hat{\text{i}}+9\hat{\text{j}}$
- $9\hat{\text{i}}+8\hat{\text{j}}$
- Find the vector $\overline{\text{BC}}$ in terms of $\hat{\text{i}},\hat{\text{j}}.$
- $\hat{\text{i}}-2\hat{\text{j}}$
- $\hat{\text{i}}+2\hat{\text{j}}$
- $2\hat{\text{i}}+\hat{\text{j}}$
- $2\hat{\text{i}}-\hat{\text{j}}$
- Length of vector $\overline{\text{AB}}$ is:
- $\sqrt{67}\text{ units}$
- $\sqrt{85}\text{ units}$
- 90 units
- 100 units
- If $\vec{\text{M}}=4\hat{\text{j}}+3\hat{\text{k}},$ then its unit vector is:
- $\frac{4}{5}\hat{\text{j}}+\frac{3}{5}\hat{\text{k}}$
- $\frac{4}{5}\hat{\text{j}}-\frac{3}{5}\hat{\text{k}}$
- $-\frac{4}{5}\hat{\text{j}}+\frac{3}{5}\hat{\text{k}}$
- $-\frac{4}{5}\hat{\text{j}}-\frac{3}{5}\hat{\text{k}}$
View full solution →Ritika starts walking from his house to shopping mall. Instead of going to the mall directly, she first goes to a ATM, from there to her daughter's school and then reaches the mall. ln the diagram, A, B, C, and D represent the coordinates of House, ATM, School and Mall respectively.
Based on the above information, answer the following questions.
- Distance between House (A) and ATM (B) is:
- $3\text{ units}$
- $3\sqrt{2}\text{ units}$
- $\sqrt{2}\text{ units}$
- $4\sqrt{2}\text{ units}$
- Distance between ATM (B) and School (C) is:
- $\sqrt{2}\text{ units}$
- $2\sqrt{2}\text{ units}$
- $3\sqrt{2}\text{ units}$
- $4\sqrt{2}\text{ units}$
- Distance between School (C) and Shopping mall (D) is:
- $3\sqrt{2}\text{ units}$
- $5\sqrt{2}\text{ units}$
- $7\sqrt{2}\text{ units}$
- $10\sqrt{2}\text{ units}$
- What is the total distance travelled by Ritika:
- $4\sqrt{2}\text{ units}$
- $6\sqrt{2}\text{ units}$
- $8\sqrt{2}\text{ units}$
- $9\sqrt{2}\text{ units}$
- What is the extra distance travelled by Ritika in reaching the shopping mall?
- $3\sqrt{2}\text{ units}$
- $5\sqrt{2}\text{ units}$
- $6\sqrt{2}\text{ units}$
- $7\sqrt{2}\text{ units}$
View full solution →Ginni purchased an air plant holder which is in the shape of a tetrahedron.
Let A, B, C, and Dare the coordinates of the air plant holder where $\text{A}\equiv(1,1,1),\text{B}\equiv(2,1,3),\text{C}\equiv(3,2,2)$ and $\text{D}\equiv(3,3,4).$
Based on the above information, answer the following questions.
- Find the position vector of $\overline{\text{AB}}.$
- $-\hat{\text{i}}-2\hat{\text{k}}$
- $2\hat{\text{i}}+\hat{\text{k}}$
- $\hat{\text{i}}+2\hat{\text{k}}$
- $-2\hat{\text{i}}-\hat{\text{k}}$
- Find the position vector of $\overline{\text{AC}}.$
- $2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$
- $2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$
- $-2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$
- $\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}$
- Find the position vector of $\overline{\text{AD}}.$
- $2\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}}$
- $\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$
- $3\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}$
- $2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$
- Area of $\triangle\text{ABC}=$
- $\frac{\sqrt{11}}{2}\text{sq}.\text{units}$
- $\frac{\sqrt{14}}{2}\text{sq}.\text{units}$
- $\frac{\sqrt{13}}{2}\text{sq}.\text{units}$
- $\frac{\sqrt{17}}{2}\text{sq}.\text{units}$
- Find the unit vector along $\overline{\text{AD}}.$
- $\frac{1}{\sqrt{17}}(2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$
- $\frac{1}{\sqrt{17}}(3\hat{\text{i}}+3\hat{\text{j}}+2\hat{\text{k}})$
- $\frac{1}{\sqrt{11}}(2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$
- $(2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$
View full solution →A building is to be constructed in the form of a triangular pyramid, ABCD as shown in the figure.
Let its angular points are A(0, 1, 2), B(3, 0, 1), C(4, 3, 6), and D(2, 3, 2), and G be the point of intersection of the medians of $\triangle\text{BCD}.$
Based on the above information, answer the following questions.
- The coordinates of point Gare:
- (2, 3, 3)
- (3, 3, 2)
- (3, 2, 3)
- (0, 2, 3)
- The length of vector $\overline{\text{AG}}$ is:
- $\sqrt{17}\text{ units}$
- $\sqrt{11}\text{ units}$
- $\sqrt{13}\text{ units}$
- $\sqrt{19}\text{ units}$
- Area of $\triangle\text{ABC}$ (in sq. units) is:
- $\sqrt{10}$
- $2\sqrt{10}$
- $3\sqrt{10}$
- $5\sqrt{10}$
- The sum of lengths of $\overline{\text{AB}}$ and $\overline{\text{AC}}$ is:
- 5 units
- 9.32 units
- 10 units
- 11 units
- The length of the perpendicular from the vertex D on the opposite face is:
- $\frac{6}{\sqrt{10}}\text{ units}$
- $\frac{2}{\sqrt{10}}\text{ units}$
- $\frac{3}{\sqrt{10}}\text{ units}$
- $8\sqrt{10}\text{ units}$
View full solution →