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:
View full solution →Question types
Vector Algebra question types
236 questions across 9 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.
236
Questions
9
Question groups
5
Question types
01
M.C.Q (1 Marks)
65 Q→02Assertion (A) & Reason (B) MCQ
23 Q→031 Marks Question
38 Q→042 Marks Questions
50 Q→053 Marks Question
20 Q→065 Marks Questions
17 Q→07Case study (4 Marks)
11 Q→08Fill In The Blanks[1 Marks ]
8 Q→09True False[1 Marks ]
4 Q→Sample Questions
Vector 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
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
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 :
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.
View full solution →$\vec{a}=6 \hat{i}+2 \hat{j}-8 \hat{k}, \vec{b}=10 \hat{i}-2 \hat{j}-6 \hat{k}, \vec{c}=4 \hat{i}-4 \hat{j}+2 \hat{k}$
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.
Assertion (A): $(\vec{b} \cdot \vec{c}) \vec{a}$ is a scalar quantity.
Reason $(R)$ : Dot product of two vectors is a scalar quantity.
View full solution →Reason $(R)$ : Dot product of two vectors is a scalar quantity.
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 $\vec{b}, \vec{a} \times \vec{b}=\vec{b} \times \vec{a}$.
View full solution →Reason (R): For two non-zero vectors $\vec{a}$ and $\vec{b}, \vec{a} \times \vec{b}=\vec{b} \times \vec{a}$.
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.
View full solution →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.
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.
View full solution →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.
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 $P(x_1, y_1, z_1)$ and $Q (x_2, y_2, z_2)$
View full solution →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:
View full solution →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?
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.
View full solution →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:
- $\frac{4}{5}\hat{\text{j}}+\frac{3}{5}\hat{\text{k}}$
- $\frac{4}{5}\hat{\text{j}}-\frac{3}{5}\hat{\text{k}}$
- $-\frac{4}{5}\hat{\text{j}}+\frac{3}{5}\hat{\text{k}}$
- $-\frac{4}{5}\hat{\text{j}}-\frac{3}{5}\hat{\text{k}}$
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.
View full solution →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?
- $3\sqrt{2}\text{ units}$
- $5\sqrt{2}\text{ units}$
- $6\sqrt{2}\text{ units}$
- $7\sqrt{2}\text{ units}$
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.
View full solution →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 $\overline{\text{AD}}.$
- $\frac{1}{\sqrt{17}}(2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$
- $\frac{1}{\sqrt{17}}(3\hat{\text{i}}+3\hat{\text{j}}+2\hat{\text{k}})$
- $\frac{1}{\sqrt{11}}(2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$
- $(2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$
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.
View full solution →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:
- $\frac{6}{\sqrt{10}}\text{ units}$
- $\frac{2}{\sqrt{10}}\text{ units}$
- $\frac{3}{\sqrt{10}}\text{ units}$
- $8\sqrt{10}\text{ units}$
Fill in the blanks.
If $\vec{\text{a}}$ is any non-zero vector, then $(\vec{\text{a}}\cdot\vec{\text{i}})\vec{\text{i}}+(\vec{\text{a}}\cdot\vec{\text{j}})\vec{\text{j}}+(\vec{\text{a}}\cdot\vec{\text{k}})\vec{\text{k}}$ equal ________.
View full solution →If $\vec{\text{a}}$ is any non-zero vector, then $(\vec{\text{a}}\cdot\vec{\text{i}})\vec{\text{i}}+(\vec{\text{a}}\cdot\vec{\text{j}})\vec{\text{j}}+(\vec{\text{a}}\cdot\vec{\text{k}})\vec{\text{k}}$ equal ________.
Fill in the blanks.
The value of the expression $|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}\cdot\vec{\text{b}})^2$ is ________.
View full solution →The value of the expression $|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}\cdot\vec{\text{b}})^2$ is ________.
Fill in the blanks.
If $|\vec{\text{a}}\times\vec{\text{b}}|^2+|\vec{\text{a}}\cdot\vec{\text{b}}|^2=144$ and $|\vec{\text{a}}|=4,$ then $|\vec{\text{b}}|^2$ is equal to ________.
View full solution →If $|\vec{\text{a}}\times\vec{\text{b}}|^2+|\vec{\text{a}}\cdot\vec{\text{b}}|^2=144$ and $|\vec{\text{a}}|=4,$ then $|\vec{\text{b}}|^2$ is equal to ________.
Fill in the blanks.
The values of k for which $|\text{k}\vec{\text{a}}|<|\vec{\text{a}}|$ and $\text{k}\vec{\text{a}}=\frac{1}{2}\vec{{\text{a}}}$ is a parallel to $\vec{\text{a}}$ holds true are _________.
View full solution →The values of k for which $|\text{k}\vec{\text{a}}|<|\vec{\text{a}}|$ and $\text{k}\vec{\text{a}}=\frac{1}{2}\vec{{\text{a}}}$ is a parallel to $\vec{\text{a}}$ holds true are _________.
Fill in the blanks.
The vector $\vec{\text{a}}+\vec{\text{b}}$ bisects the angle between the non-collinear vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ if _________.
View full solution →The vector $\vec{\text{a}}+\vec{\text{b}}$ bisects the angle between the non-collinear vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ if _________.
Two collinear vectors having the same magnitude are equal.
Two vectors having same magnitude are collinear.
Two collinear vectors are always equal in magnitude.
$\vec a $ and $-\vec a$ are collinear.
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