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Mathematics · Class 12 Science

Ch 8Vectors — Class 12 Mathematics, concept-first.

Physical quantities are broadly of two kinds. A scalar quantity is completely described by a single real number together with an appropriate unit -- examples are length, mass, temperature, time and speed.

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Key concepts

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Scalar (Dot) Product

For non-zero vectors with included angle (), the scalar (dot) product is the number

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In previous exams

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WBJEE

1 concept · 1 appearance

2025

Chapter contents

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

1

Vectors and Scalars — Magnitude and Direction

Physical quantities are broadly of two kinds. A scalar quantity is completely described by a single real number together with an appropriate unit -- examples are length, mass, temperature, time and sp…

2

Direction Cosines and Direction Ratios

Let a vector (or the directed line from the origin to a point ) make angles with the positive -, - and -axes respectively.

3

Types of Vectors

Vectors are classified by special relationships of magnitude, direction, or position. These classifications are used constantly throughout vector algebra.

4

Position Vector and Components of a Vector

Fix an origin . For any point in space, the vector is called the position vector of relative to :

5

Addition of Vectors and Scalar Multiplication

If two vectors and are placed so the terminal point of the first coincides with the initial point of the second, their sum is the vector from the initial point of the first to the terminal point of th…

6

Position Vector of a Point Dividing a Segment (Section Formula)

Let and be two points with position vectors and (relative to a fixed origin ), and let be the point on segment that divides it internally in the ratio (so , with between and ).

7

Scalar (Dot) Product of Vectors

For two nonzero vectors and with the angle between them (where , measured with both vectors drawn from a common initial point), the scalar product (or dot product) is the real number

8

Vector (Cross) Product of Vectors

For two nonzero, non-parallel vectors and with angle between them (), the vector product (or cross product) is itself a vector, defined by

Summary

- A vector has both magnitude and direction; a scalar has magnitude only. - Direction cosines are the cosines of the angles a vector makes with the axes, and always satisfy ; direction ratios are any…

Sample & Board Papers

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

+Show 20 questions20 questions
  1. Q1Show that A(2, 3, -4), B(1, -2, 3) and C(3, 8, -11) are collinear.Preview
  2. Q2If vector a = 5i - j - 3k and vector b = i + 3j - 5k, show that (a + b) and (a - b) are perpendicular to each other.Preview
  3. Q3vector a, vector b, vector c be three vectors such that a + b + c = 0 and |a| = 1, |b| = 4, |c| = 2. Evaluate a.b + b.c + c.a.Preview
  4. Q4If sum of two unit vectors be a unit vector, then show that difference of those two vectors is sqrt(3).Preview
  5. Q5vector a = i + 3j - k and vector b = 2i + 6j + lambda k. If vector a and vector b are parallel, then the value of lambda is (a) 3 (b) -6 (c)…Preview
  6. Q6If vector a = 5î - ĵ - 3k̂ and vector b = î + 3ĵ - 5k̂, then show that (a + b) and (a - b) are perpendicular to each other.Preview
  7. Q7If sum of two unit vectors be a unit vector, then show that difference of those two vectors is √3.Preview
  8. Q8Find the unit vector in the direction of vector a = 2î + 3ĵ + √3k̂.Preview
  9. Q9a, b, c three vectors are such that |a|=3, |b|=4, |c|=5 and each vector is perpendicular to the sum of other two vectors. Find |a+b+c|.Preview
  10. Q10If a = î+ĵ+k̂, b = ĵ-k̂, find vector c such that a×c = b and a.c = 3.Preview
  11. Q11For vectors a⃗ and b⃗, |a⃗| = √3, |b⃗| = 2 and a⃗.b⃗ = √6, angle between a⃗ and b⃗ is (a) π/2 (b) π/6 (c) π/3 (d) π/4Preview
  12. Q12If a⃗ = 3î - 2ĵ + k̂ and b⃗ = î - 3ĵ + 4k̂, find the area of the parallelogram whose adjacent sides are a⃗ and b⃗.Preview
  13. Q13If the vectors a⃗ = 2î + 2ĵ + 3k̂, b⃗ = -î + 2ĵ + k̂ and c⃗ = 3î + ĵ are such that a⃗ + λb⃗ and c⃗ are perpendicular to each other, find the…Preview
  14. Q14Show that (a⃗+b⃗).{(b⃗+c⃗)×(c⃗+a⃗)} = 2a⃗(b⃗+c⃗).Preview
  15. Q15If |a⃗|=4, |b⃗|=2√3, |a⃗×b⃗|=12 then the angle between the vectors a⃗ and b⃗ is (a) π/3 (b) π/6 (c) π/4 (d) π/2Preview
  16. Q16If ABCDEF is a regular hexagon, then prove that AD⃗ + EB⃗ + FC⃗ = 4AB⃗.Preview
  17. Q17If G is the centroid of the triangle ABC, then prove by the vector method that GA⃗ + GB⃗ + GC⃗ = 0⃗.Preview
  18. Q18If α⃗, β⃗, γ⃗ be the unit vectors satisfying the condition α⃗+β⃗+γ⃗=0, then show that α⃗.β⃗+β⃗.γ⃗+γ⃗.α⃗ = -3/2. Hence examine whether the ve…Preview
  19. Q19If the vectors $\vec{\alpha} = a\hat{i} + a\hat{j} + c\hat{k}$; $\vec{\beta} = \hat{i} + \hat{k}$; $\vec{\gamma} = c\hat{i} + c\hat{j} + b\h…Preview
  20. Q20The adjacent sides of a parallelogram are represented by the vectors $\hat{i}$ and $\hat{i} + \hat{j}$. Determine the area of the parallelog…Preview

More questions

31 Q
+Show 2 questions2 questions
  1. Q30If $\vec a,\vec b,\vec c$ are vectors such that $\vec a+\vec b+\vec c=\vec 0$, and $|\vec a|=3,\ |\vec b|=5,\ |\vec c|=7$, find the angle be…Free
  2. Q31Using vectors, show that the points $A(1,2,7)$, $B(2,6,3)$ and $C(3,10,-1)$ are collinear.Preview
+Show 9 questions9 questions
  1. Example 1Find the magnitude and direction cosines of the vector $\vec{PQ}$, where $P(2,1,-1)$ and $Q(5,5,11)$.Free
  2. Example 2If a vector has direction ratios $-1,2,2$, find its direction cosines. Hence find the angle it makes with the $y$-axis.Free
  3. Example 3Show that the vectors $\vec a=2\hat i-3\hat j+4\hat k$ and $\vec b=-4\hat i+6\hat j-8\hat k$ are parallel, and state whether they point in t…Free
  4. Example 4Find a unit vector in the direction of $\vec{AB}$, where $A(1,2,3)$ and $B(4,6,15)$.Preview
  5. Example 5If $\vec a=\hat i+2\hat j-\hat k$ and $\vec b=2\hat i-\hat j+3\hat k$, find $2\vec a-3\vec b$ and its magnitude.Preview
  6. Example 6Find the position vector of the point $R$ which divides the line segment joining $A(2,3,4)$ and $B(4,5,6)$ internally in the ratio $2:3$.Preview
  7. Example 7Find the angle between the vectors $\vec a=3\hat i-\hat j+2\hat k$ and $\vec b=\hat i-\hat j-\hat k$.Preview
  8. Example 8Find the projection of $\vec a=2\hat i+3\hat j-\hat k$ on $\vec b=\hat i-2\hat j+2\hat k$.Preview
  9. Example 9Find the area of the triangle with vertices $A(1,1,1)$, $B(1,2,3)$ and $C(2,3,1)$, using the vector (cross) product.Preview
+Show 4 questions4 questions
  1. Q10Find the magnitude and direction cosines of the vector $4\hat i-4\hat j+7\hat k$.Free
  2. Q11Find the direction cosines of the vector $\vec{AB}$ joining the points $A(-2,4,-5)$ and $B(1,2,3)$.Free
  3. Q12A vector makes angles of $90°$ and $60°$ with the $x$-axis and $y$-axis respectively. Find the angle it makes with the $z$-axis (take the ac…Preview
  4. Q13A vector has direction ratios $2,-1,-2$. Find its direction cosines and verify that $l^2+m^2+n^2=1$.Preview
+Show 3 questions3 questions
  1. Q14Classify each of the following as a scalar or a vector quantity, and state which of the vectors (if any) is a unit vector: (i) $5$ kg (ii) $…Free
  2. Q15Using position vectors, show that the points $A(2,-1,3)$, $B(4,3,-1)$ and $C(3,1,1)$ are collinear.Preview
  3. Q16Given that $\vec a=-2\hat i+3\hat j+5\hat k$ and $\vec b=6\hat i-9\hat j-15\hat k$ are collinear vectors, find the scalar $\lambda$ such tha…Preview
+Show 3 questions3 questions
  1. Q17If $\vec a=3\hat i-2\hat j+\hat k$ and $\vec b=2\hat i+\hat j-3\hat k$, find $\vec a+\vec b$ and $\vec a-\vec b$.Free
  2. Q18Find a vector in the direction of $\vec a=\hat i-2\hat j$ that has magnitude $7$.Preview
  3. Q19If $\vec a=2\hat i+3\hat j-\hat k$, $\vec b=-\hat i+2\hat j-4\hat k$ and $\vec c=\hat i-\hat j+\hat k$, find $\vec a+2\vec b-3\vec c$.Preview
+Show 3 questions3 questions
  1. Q20Find the position vector of the midpoint of the line segment joining $A(3,-2,1)$ and $B(1,4,-3)$.Free
  2. Q21Find the position vector of the point which divides the line segment joining $A(1,-2,1)$ and $B(4,7,-2)$ internally in the ratio $2:1$.Preview
  3. Q22Find the position vector of the point which divides the line segment joining $A(2,1,-3)$ and $B(4,3,-5)$ externally in the ratio $3:2$.Preview
+Show 4 questions4 questions
  1. Q23If $\vec a=\hat i+\hat j+\hat k$ and $\vec b=\hat i-\hat j+2\hat k$, find $\vec a\cdot\vec b$ and the angle between $\vec a$ and $\vec b$.Free
  2. Q24For what value of $\lambda$ are the vectors $\vec a=2\hat i+\lambda\hat j+\hat k$ and $\vec b=\hat i-2\hat j+3\hat k$ perpendicular?Free
  3. Q25If $|\vec b|=3$ and $(\vec a+\vec b)\cdot(\vec a-\vec b)=8$, find $|\vec a|$.Preview
  4. Q26Find the projection of $\vec b=2\hat i+3\hat j+2\hat k$ on $\vec a=\hat i+2\hat j+\hat k$.Preview
+Show 3 questions3 questions
  1. Q27If $\vec a=2\hat i-3\hat j+\hat k$ and $\vec b=\hat i+4\hat j-2\hat k$, find $\vec a\times\vec b$.Free
  2. Q28Find a unit vector perpendicular to both $\vec a=\hat i-\hat j+\hat k$ and $\vec b=2\hat i+\hat j-2\hat k$.Preview
  3. Q29Find the area of the parallelogram whose adjacent sides are represented by $\vec a=3\hat i+\hat j-2\hat k$ and $\vec b=\hat i-3\hat j+4\hat…Preview