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Mathematics · Class 11 Science

Ch 8Vector Algebra-I — Class 11 Mathematics, concept-first.

A pilot planning a flight has to head the aircraft into the wind at just the right angle so that the wind's own push is exactly counteracted and the plane still reaches its destination — done either with a navigation computer or, in its absence, by hand with a working knowledge of vectors.

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Key concepts

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Scalars and Vectors; Types of Vectors

A scalar is completely fixed once you know its size — distance, mass, speed, temperature. A vector needs both a magnitude and a direction — force, displacement, velocity.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

8.1

Introduction

A pilot planning a flight has to head the aircraft into the wind at just the right angle so that the wind's own push is exactly counteracted and the plane still reaches its destination — done either w…

8.2

Scalars and Vectors

Scalars vs vectors. A scalar is a quantity that is completely described once you know how much of it there is — its magnitude, and nothing else.

8.3

Representation of a Vector and Types of Vectors

Tail, tip and support. Every vector has a tail (its initial point ) and a tip (its terminal point ). The magnitude is the length of the segment .

8.4

Algebra of Vectors

Just as we add, subtract and scale ordinary numbers, and did the same for matrices, we now define these three operations for vectors: addition of two vectors, subtraction of one vector from another, a…

8.4.1

Addition of Vectors

Motivating the definition. Imagine a unit mass at the origin, pushed by two unit forces (along the positive -axis) and (along the positive -axis), acting one after the other: first moves the object fr…

8.4.2

Difference between Two Vectors

The reverse of a vector. For a vector , its reverse, written , is the vector with the same magnitude as but the opposite direction. If , then .

8.4.3

Scalar Multiplication of a Vector

Definition. For a vector and a scalar (real number) , the vector is called the scalar multiple of by .

8.4.4

Some Properties and Results

For any vectors and scalars , scalar multiplication obeys the same rules as ordinary numbers:

8.5

Position Vectors

Position vectors. Once we fix an origin , every point in the plane or in space can be represented by a single vector: the vector is called the position vector of with respect to .

+Exercise 8.1i12 questions
  1. Q1Represent graphically the displacement of (i) $45\text{ cm},\ 30^\circ$ north of east (ii) $80\text{ km},\ 60^\circ$ south of west.Free
  2. Q2Prove that the relation $R$ defined on the set $V$ of all vectors by '$\vec a\,R\,\vec b$ if $\vec a=\vec b$' is an equivalence relation on…Free
  3. Q3Let $\vec a$ and $\vec b$ be the position vectors of the points $A$ and $B$. Prove that the position vectors of the points which trisect the…Free
  4. Q4If $D$ and $E$ are the midpoints of the sides $AB$ and $AC$ of a triangle $ABC$, prove that $\vec{BE}+\vec{DC}=\dfrac{3}{2}\vec{BC}$.Preview
  5. Q5Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is half the lengt…Preview
  6. Q6Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.Preview
  7. Q7If $\vec a$ and $\vec b$ represent a side and a diagonal of a parallelogram, find the other sides and the other diagonal.Preview
  8. Q8If $\vec{PO}+\vec{OQ}=\vec{QO}+\vec{OR}$, prove that the points $P,Q,R$ are collinear.Preview
  9. Q9If $D$ is the midpoint of the side $BC$ of a triangle $ABC$, prove that $\vec{AB}+\vec{AC}=2\vec{AD}$.Preview
  10. Q10If $G$ is the centroid of a triangle $ABC$, prove that $\vec{GA}+\vec{GB}+\vec{GC}=\vec 0$.Preview
  11. Q11Let $A,B,C$ be the vertices of a triangle. Let $D,E,F$ be the midpoints of the sides $BC,CA,AB$ respectively. Show that $\vec{AD}+\vec{BE}+\…Preview
  12. Q12If $ABCD$ is a quadrilateral and $E$ and $F$ are the midpoints of $AC$ and $BD$ respectively, then prove that $\vec{AB}+\vec{AD}+\vec{CB}+\v…Preview
8.6

Resolution of Vectors

So far we've described vectors abstractly, as arrows with a magnitude and a direction. To actually compute with vectors — add them numerically, find their magnitudes, and so on — it helps enormously t…

8.6.1

Resolution of a Vector in Two Dimensions

Setting up. Let be the unit vectors along the positive -axis and positive -axis respectively, both with initial point at the origin . Let be any point in the plane, so is its position vector.

8.6.2

Resolution of a Vector in Three Dimensions

Setting up in space. Let be unit vectors along the positive axes, all with initial point at the origin . Let be any point in space.

8.6.3

Matrix Representation of a Vector

Vectors as matrices. A vector with three rectangular components can equally well be written as a row matrix or a column matrix . Concretely, for :

8.7

Direction Cosines and Direction Ratios

Direction angles. Let be a point in space at distance from the origin , and let be the feet of the perpendiculars from to the axes.

+Exercise 8.2i17 questions
  1. Q1Verify whether the following ratios are direction cosines of some vector or not. (i) $\dfrac15,\dfrac35,\dfrac45$ (ii) $\dfrac12,\dfrac12,\d…Free
  2. Q2Find the direction cosines of a vector whose direction ratios are (i) $1,2,3$ (ii) $3,-1,3$ (iii) $0,0,7$Free
  3. Q3Find the direction cosines and direction ratios for the following vectors. (i) $3\hat i-4\hat j+8\hat k$ (ii) $3\hat i+\hat j+\hat k$ (iii)…Free
  4. Q4A triangle is formed by joining the points $(1,0,0),(0,1,0)$ and $(0,0,1)$. Find the direction cosines of the medians.Preview
  5. Q5If $\dfrac12,\dfrac12,a$ are the direction cosines of some vector, then find $a$.Preview
  6. Q6If $(a,\,a+b,\,a+b+c)$ is one set of direction ratios of the line joining $(1,0,0)$ and $(0,1,0)$, then find a set of values of $a,b,c$.Preview
  7. Q7Show that the vectors $2\hat i - \hat j + \hat k,\ 3\hat i - 4\hat j - 4\hat k,\ \hat i - 3\hat j - 5\hat k$ form a right angled triangle.Preview
  8. Q8Find the value of $\lambda$ for which the vectors $\vec a=3\hat i+2\hat j+9\hat k$ and $\vec b=\hat i+\lambda\hat j+3\hat k$ are parallel.Preview
  9. Q9Show that the following vectors are coplanar (i) $\hat i - 2\hat j + 3\hat k,\ -2\hat i + 3\hat j - 4\hat k,\ -\hat j + 2\hat k$ (ii) $5\hat…Preview
  10. Q10Show that the points whose position vectors $4\hat i+5\hat j+\hat k,\ -\hat j-\hat k,\ 3\hat i+9\hat j+4\hat k$ and $-4\hat i+4\hat j+4\hat…Preview
  11. Q11If $\vec a=2\hat i+3\hat j-4\hat k$, $\vec b=3\hat i-4\hat j-5\hat k$, and $\vec c=-3\hat i+2\hat j+3\hat k$, find the magnitude and directi…Preview
  12. Q12The position vectors of the vertices of a triangle are $\hat i-2\hat j+3\hat k$; $3\hat i+4\hat j-5\hat k$ and $2\hat i+3\hat j-7\hat k$. Fi…Preview
  13. Q13Find the unit vector parallel to $3\vec a-2\vec b+4\vec c$ if $\vec a=3\hat i-\hat j-4\hat k,\ \vec b=-2\hat i+4\hat j-3\hat k$, and $\vec c…Preview
  14. Q14The position vectors $\vec a,\vec b,\vec c$ of three points satisfy the relation $2\vec a-7\vec b+5\vec c=\vec 0$. Are these points collinea…Preview
  15. Q15The position vectors of the points $P,Q,R,S$ are $\hat i+\hat j+\hat k,\ 2\hat i+5\hat j,\ 3\hat i+2\hat j-3\hat k$, and $\hat i-6\hat j-\ha…Preview
  16. Q16Find the value or values of $m$ for which $m(\hat i+\hat j+\hat k)$ is a unit vector.Preview
  17. Q17Show that the points $A(1,1,1)$, $B(1,2,3)$ and $C(2,-1,1)$ are vertices of an isosceles triangle.Preview
8.8

Product of Vectors

We have now defined addition, subtraction and scalar multiplication of vectors. The remaining natural operation is multiplying two vectors together — but here vectors differ sharply from ordinary numb…

8.8.1

Angle between Two Vectors

Defining the angle. Let and be any two vectors, brought to a common initial point . The angle between and is the angle between their directions at that common point — whether the two arrows visually c…

8.8.2

Scalar Product

Definition. Let be two non-zero vectors with included angle . Their scalar product (or dot product), written , is defined as the number Because the result of is a scalar, this is called the scalar pro…

8.8.3

Properties of Scalar Product

(i) Commutative. .

+Exercise 8.3i14 questions
  1. Q1Find $\vec a\cdot\vec b$ when (i) $\vec a=\hat i-2\hat j+\hat k$ and $\vec b=3\hat i-4\hat j-2\hat k$ (ii) $\vec a=2\hat i+2\hat j-\hat k$ a…Free
  2. Q2Find the value of $\lambda$ for which the vectors $\vec a$ and $\vec b$ are perpendicular, where (i) $\vec a=2\hat i+\hat j+\lambda\hat k$ a…Free
  3. Q3If $\vec a$ and $\vec b$ are two vectors such that $|\vec a|=10,|\vec b|=15$ and $\vec a\cdot\vec b=75\sqrt2$, find the angle between $\vec…Free
  4. Q4Find the angle between the vectors (i) $2\hat i+3\hat j-6\hat k$ and $6\hat i-3\hat j+2\hat k$ (ii) $\hat i-\hat j$ and $\hat j-\hat k$.Preview
  5. Q5If $\vec a,\vec b,\vec c$ are three vectors such that $2\vec a+\vec b+\vec c=\vec 0$ and $|\vec a|=3,|\vec b|=4,|\vec c|=7$, find the angle…Preview
  6. Q6Show that the vectors $\vec a=2\hat i+3\hat j+6\hat k$, $\vec b=6\hat i+2\hat j-3\hat k$, and $\vec c=3\hat i-6\hat j+2\hat k$ are mutually…Preview
  7. Q7Show that the vectors $-\hat i-2\hat j-6\hat k,\ 2\hat i-\hat j+\hat k$, and $-\hat i+3\hat j+5\hat k$ form a right angled triangle.Preview
  8. Q8If $|\vec a|=5,|\vec b|=6,|\vec c|=7$ and $\vec a+\vec b+\vec c=\vec 0$, find $\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a$.Preview
  9. Q9Show that the points $(2,-1,3)$, $(4,3,1)$ and $(3,1,2)$ are collinear.Preview
  10. Q10If $\vec a,\vec b$ are unit vectors and $\theta$ is the angle between them, show that (i) $\sin\dfrac\theta2=\dfrac12|\vec a-\vec b|$ (ii) $…Preview
  11. Q11Let $\vec a,\vec b,\vec c$ be three vectors such that $|\vec a|=3,|\vec b|=4,|\vec c|=5$ and each one of them is perpendicular to the sum of…Preview
  12. Q12Find the projection of the vector $3\hat i+7\hat j+\hat k$ on the vector $2\hat i+6\hat j+3\hat k$.Preview
  13. Q13Find $\lambda$, when the projection of $4\hat i+\hat j+\lambda\hat k$ on $2\hat i+6\hat j+3\hat k$ is $4$ units.Preview
  14. Q14Three vectors $\vec a,\vec b,\vec c$ are such that $|\vec a|=2,|\vec b|=3,|\vec c|=4$, and $\vec a+\vec b+\vec c=\vec 0$. Find $4\,\vec a\cd…Preview
8.8.4

Vector Product

Right-handed and left-handed systems. If you align the fingers of your right hand along and curl them towards (through the angle less than ), your thumb points along ; curling the other way (from towa…

8.8.5

Properties of Vector Product

(i) Anti-commutative (not commutative). By definition, , since (rather than ) form a right-handed system. So The vector product is genuinely non-commutative.

+Exercise 8.4i10 questions
  1. Q1Find the magnitude of $\vec a\times\vec b$ if $\vec a=2\hat i+3\hat j+\hat k$ and $\vec b=3\hat i+5\hat j-2\hat k$.Free
  2. Q2Show that $\vec a\times(\vec b+\vec c)+\vec b\times(\vec c+\vec a)+\vec c\times(\vec a+\vec b)=\vec 0$.Free
  3. Q3Find the vectors of magnitude $10\sqrt3$ that are perpendicular to the plane which contains $2\hat i+\hat j+\hat k$ and $3\hat i+4\hat j+\ha…Free
  4. Q4Find the unit vectors perpendicular to each of the vectors $\vec a+\vec b$ and $\vec a-\vec b$, where $\vec a=\hat i+\hat j+\hat k$ and $\ve…Preview
  5. Q5Find the area of the parallelogram whose two adjacent sides are determined by the vectors $2\hat i+3\hat j+\hat k$ and $3\hat i-2\hat j+\hat…Preview
  6. Q6Find the area of the triangle whose vertices are $A(3,-1,2)$, $B(1,-1,-3)$ and $C(4,-3,1)$.Preview
  7. Q7If $\vec a,\vec b,\vec c$ are position vectors of the vertices $A,B,C$ of a triangle $ABC$, show that the area of the triangle $ABC$ is $\df…Preview
  8. Q8For any vector $\vec a$ prove that $|\vec a\times\hat i|^2+|\vec a\times\hat j|^2+|\vec a\times\hat k|^2=2|\vec a|^2$.Preview
  9. Q9Let $\vec a,\vec b,\vec c$ be unit vectors such that $\vec a\cdot\vec b=0=\vec a\cdot\vec c$ and the angle between $\vec b$ and $\vec c$ is…Preview
  10. Q10Find the angle between the vectors $2\hat i+\hat j-\hat k$ and $2\hat i+\hat j+\hat k$ using vector product.Preview
8.9

Chapter Review

This chapter developed vector algebra from the ground up: what a vector is and how it differs from a scalar; how to add, subtract, and scale vectors both geometrically (triangle/parallelogram laws) an…

+Exercise 8.5i25 questions
  1. Q1The value of $\vec{AB}+\vec{BC}+\vec{DA}+\vec{CD}$ is (1) $\vec{AD}$ (2) $\vec{CA}$ (3) $\vec 0$ (4) $-\vec{AD}$Free
  2. Q2If $\vec a+2\vec b$ and $3\vec a+m\vec b$ are parallel, then the value of $m$ is (1) $3$ (2) $\dfrac13$ (3) $6$ (4) $\dfrac16$Free
  3. Q3The unit vector parallel to the resultant of the vectors $\hat i+\hat j-\hat k$ and $\hat i-2\hat j+\hat k$ is (1) $\dfrac{-\hat i+\hat j+\h…Free
  4. Q4A vector $\vec{OP}$ makes $60^\circ$ and $45^\circ$ with the positive direction of the $x$ and $y$ axes respectively. Then the angle it make…Preview
  5. Q5If $\vec{BA}=3\hat i+2\hat j+\hat k$ and the position vector of $B$ is $\hat i+3\hat j-\hat k$, then the position vector of $A$ is (1) $4\ha…Preview
  6. Q6A vector makes equal angles with the positive directions of the coordinate axes. Then each angle is equal to (1) $\cos^{-1}\!\left(\dfrac1{\…Preview
  7. Q7The vectors $\vec a-\vec b$, $\vec b-\vec c$, $\vec c-\vec a$ are (1) parallel to each other (2) unit vectors (3) mutually perpendicular vec…Preview
  8. Q8If $ABCD$ is a parallelogram, then $\vec{AB}+\vec{AD}+\vec{CB}+\vec{CD}$ is equal to (1) $2(\vec{AB}+\vec{AD})$ (2) $4\vec{AC}$ (3) $4\vec{B…Preview
  9. Q9One of the diagonals of parallelogram $ABCD$ with $\vec a$ and $\vec b$ as adjacent sides is $\vec a+\vec b$. The other diagonal $\vec{BD}$…Preview
  10. Q10If $\vec a,\vec b$ are the position vectors $A$ and $B$, then which one of the following points whose position vector lies on $AB$, is (1) $…Preview
  11. Q11If $\vec a,\vec b,\vec c$ are the position vectors of three collinear points, then which of the following is true? (1) $\vec a=\vec b+\vec c…Preview
  12. Q12If $9\vec a+7\vec b=16\vec r$, then the point $P$ whose position vector is $\vec r$ divides the line joining the points with position vector…Preview
  13. Q13If $\lambda\hat i+2\lambda\hat j+2\lambda\hat k$ is a unit vector, then the value of $\lambda$ is (1) $\dfrac13$ (2) $\dfrac14$ (3) $\dfrac1…Preview
  14. Q14Two vertices of a triangle have position vectors $3\hat i+4\hat j-4\hat k$ and $2\hat i+3\hat j+4\hat k$. If the position vector of the cent…Preview
  15. Q15If $|\vec a+\vec b|=60$, $|\vec a-\vec b|=40$ and $|\vec b|=46$, then $|\vec a|$ is (1) $42$ (2) $12$ (3) $22$ (4) $32$Preview
  16. Q16If $\vec a$ and $\vec b$ have the same magnitude and the angle between them is $60^\circ$ and their scalar product is $\dfrac12$, then $|\ve…Preview
  17. Q17The value of $\theta\in\left(0,\dfrac\pi2\right)$ for which the vectors $\vec a=(\sin\theta)\hat i+(\cos\theta)\hat j$ and $\vec b=-\sqrt3\h…Preview
  18. Q18If $|\vec a|=13,|\vec b|=5$ and $\vec a\cdot\vec b=60$, then $|\vec a\times\vec b|$ is (1) $15$ (2) $35$ (3) $45$ (4) $25$Preview
  19. Q19Vectors $\vec a$ and $\vec b$ are inclined at an angle $\theta=120^\circ$. If $|\vec a|=1,|\vec b|=2$, then $\left|(\vec a+3\vec b)\times(3\…Preview
  20. Q20If $\vec a$ and $\vec b$ are two vectors of magnitude $2$ each, inclined at an angle $60^\circ$, then the angle between $\vec a$ and $\vec a…Preview
  21. Q21If the projection of $5\hat i-3\hat j-\hat k$ on the vector $\lambda\hat i+3\hat j+\hat k$ is the same as the projection of $\lambda\hat i+3…Preview
  22. Q22If $(1, 2, 4)$ and $(2, -3\lambda, -3)$ are the initial and terminal points of the vector $\hat i+5\hat j-7\hat k$, then the value of $\lamb…Preview
  23. Q23If the points whose position vectors are $10\hat i+3\hat j$, $12\hat i-5\hat j$ and $a\hat i+11\hat j$ are collinear, then $a$ is equal to (…Preview
  24. Q24If $\vec a=\hat i+\hat j+\hat k,\ \vec b=2\hat i+x\hat j+\hat k,\ \vec c=\hat i-\hat j+4\hat k$ and $\vec a\cdot(\vec b\times\vec c)=70$, th…Preview
  25. Q25If $\vec a=2\hat i+2\hat j+\hat k$, $|\vec b|=5$ and the angle between $\vec a$ and $\vec b$ is $\dfrac\pi6$, then the area of the triangle…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 35 questions35 questions
  1. Q1If $\vec{a}$ is a non-zero vector and k is a scalar such that $|k\vec{a}| = 1$ then k is equal to: (a) $\dfrac{1}{|\vec{a}|}$ (b) $|\vec{a}|…Preview
  2. Q2If $\vec{a}, \vec{b}, \vec{c}$ be the vectors represented by the three sides of a triangle, taken in order, then prove that $\vec{a} + \vec{…Preview
  3. Q3Find the vectors of magnitude 5 units, which are parallel to the vector $2\vec{i} - \vec{j}$.Preview
  4. Q4The unit vector parallel to the resultant of the vectors $\hat{i}+\hat{j}-\hat{k}$ and $\hat{i}-2\hat{j}+\hat{k}$ is: (a) $\dfrac{2\hat{i}-\…Preview
  5. Q5If $\vec{a}, \vec{b}$ are the position vectors of A and B, then which one of the following points whose position vector lies on AB? (a) $\df…Preview
  6. Q6Find a unit vector along the direction of the vector $5\hat{i}-3\hat{j}+4\hat{k}$.Preview
  7. Q7If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $\vec{a}+2\vec{b}+\vec{c}=\vec{0}$ and $|\vec{a}|=3$, $|\vec{b}|=4$, $|\vec{c}|=7…Preview
  8. Q8The unit vector parallel to the resultant of the vectors $\hat{i} + \hat{j} - \hat{k}$ and $\hat{i} - 2\hat{j} + \hat{k}$ is: (a) $\dfrac{\h…Preview
  9. Q9If $|\vec{a}| = 13$, $|\vec{b}| = 5$ and $\vec{a} \cdot \vec{b} = 60°$, then $|\vec{a} \times \vec{b}|$ is: (a) 15 (b) 35 (c) 45 (d) 25Preview
  10. Q10If G is the centroid of a triangle ABC, prove that $\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} = \vec{0}$.Preview
  11. Q11Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} = 0$ and the angle between $\vec{b}…Preview
  12. Q12Two vertices of a triangle have position vectors $3\hat{i}+4\hat{j}-4\hat{k}$ and $2\hat{i}+3\hat{j}+4\hat{k}$. If the position vector of th…Preview
  13. Q13One of the diagonals of parallelogram ABCD with $\vec{a}$ and $\vec{b}$ as adjacent sides is $\vec{a} + \vec{b}$. The other diagonal $\overr…Preview
  14. Q14If the points whose position vectors are $10\hat{i}+3\hat{j}$, $12\hat{i}-5\hat{j}$ and $a\hat{i}+11\hat{j}$ are collinear then '$a$' is equ…Preview
  15. Q15Find a unit vector along the direction of the vector $5\hat{i}-3\hat{j}+4\hat{k}$.Preview
  16. Q16Find the angle between the vectors $5\hat{i}+3\hat{j}+4\hat{k}$ and $6\hat{i}-8\hat{j}-\hat{k}$.Preview
  17. Q17(a) Prove that the points whose position vectors are $2\hat{i}+4\hat{j}+3\hat{k}$, $4\hat{i}+\hat{j}+9\hat{k}$ and $10\hat{i}-\hat{j}+6\hat{…Preview
  18. Q18The value of $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{DA} + \overrightarrow{CD}$ is: (a) $\vec{0}$ (b) $\overrightarrow{…Preview
  19. Q19If $|\vec a| = 13$, $|\vec b| = 5$ and $\vec a \cdot \vec b = 60$ then $|\vec a \times \vec b|$ is: (a) 45 (b) 15 (c) 25 (d) 35Preview
  20. Q20Find the area of the parallelogram whose adjacent sides are $\vec a = 3\hat i + \hat j + 4\hat k$ and $\vec b = \hat i - \hat j + \hat k$.Preview
  21. Q21Show that the points whose position vectors $4\hat i + 5\hat j + \hat k$, $-\hat j - \hat k$, $3\hat i + 9\hat j + 4\hat k$ and $-4\hat i +…Preview
  22. Q22If $\vec a=\hat i+\hat j+\hat k$, $\vec b=2\hat i+x\hat j+\hat k$, $\vec c=\hat i-\hat j+4\hat k$ and $\vec a\cdot(\vec b\times\vec c)=70$ t…Preview
  23. Q23If $\overrightarrow{BA}=3\hat i+2\hat j+\hat k$ and the position vector of B is $\hat i+3\hat j-\hat k$, then the position vector A is: (a)…Preview
  24. Q24Find $|\vec a\times\vec b|$, where $\vec a=3\hat i+4\hat j$ and $\vec b=\hat i+\hat j+\hat k$.Preview
  25. Q25Show that the vectors $2\hat i-\hat j+\hat k$, $3\hat i-4\hat j-4\hat k$, $\hat i-3\hat j-5\hat k$ form a right angled triangle.Preview
  26. Q26(a) If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then prove that $\overrightarrow{AB}+\overrightarrow…Preview
  27. Q27If $|\vec{a}| = 3, |\vec{b}| = 4, |\vec{c}| = 5$ and $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ then the angle between $\vec{a}$ and $\vec{b}$…Preview
  28. Q28If $\vec{a} + 2\vec{b}$ and $3\vec{a} + m\vec{b}$ are parallel, then the value of $m$ is: (a) $6$ (b) $3$ (c) $\dfrac{1}{6}$ (d) $\dfrac{1}{…Preview
  29. Q29If $\overrightarrow{PO} + \overrightarrow{OQ} = \overrightarrow{QO} + \overrightarrow{OR}$, prove that the points P, Q, R are collinear.Preview
  30. Q30Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors such that $\vec{a}.\vec{b} = \vec{a}.\vec{c} = 0$ and the angle between $\vec{b}$ and $\vec{…Preview
  31. Q31The value of $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}$ is: (a) $\vec{0}$ (b) $\overrightarrow{AD}$ (…Preview
  32. Q32If $|\vec{a}| = 13$, $|\vec{b}| = 5$ and $\vec{a}\cdot\vec{b} = 60^\circ$, then $|\vec{a}\times\vec{b}|$ is: (a) 45 (b) 15 (c) 25 (d) 35Preview
  33. Q33Find $|\vec{a}\times\vec{b}|$, where $\vec{a} = 3\hat{i}+4\hat{j}$ and $\vec{b} = \hat{i}+\hat{j}+\hat{k}$Preview
  34. Q34Show that the vectors $-\hat{i}-2\hat{j}-6\hat{k}$, $2\hat{i}-\hat{j}+\hat{k}$ and $-\hat{i}+3\hat{j}+5\hat{k}$ form a right angled triangle…Preview
  35. Q35If $ABCD$ is a quadrilateral and $E$ and $F$ are the midpoints of $AC$ and $BD$ respectively, then prove that $\overrightarrow{AB}+\overrigh…Preview