Class 12 Maths · Chapter 10
Unit: Vectors and Three-Dimensional Geometry
Tentative Study Share: 7.0 Marks
Chapter 10: Vector Algebra — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Vector Algebra.
NCERT Exercises in Chapter 10: Vector Algebra
📘 NCERT Concept Notes — Vector Algebra
01 · An arrow has two jobs
A vector carries magnitude and direction. Equal free vectors can begin at different points; moving an arrow without rotating or resizing it does not change the vector. The zero vector has magnitude zero and no definite direction.
02 · Follow the journey
Addition joins successive displacements. Components add coordinate by coordinate. To travel from P to Q, subtract the position of P from the position of Q. Scalar multiplication changes length by the absolute scalar and reverses direction for a negative scalar.
03 · Locate a point between two others
Internal and external division use the same endpoint position vectors but different ratio equations. Always write what m:n means before assigning weights.
04 · Measure alignment
Dot product produces a scalar. For nonzero vectors it is positive for an acute angle, zero for a right angle and negative for an obtuse angle. Projection measures the component along one chosen direction.
05 · Measure a plane’s spread
Cross product produces a normal vector. Order controls its direction; its magnitude gives the parallelogram area. For nonzero vectors, a zero cross product tests collinearity.
⚡ Key Formulas & Results — Vector Algebra
Components and magnitude
How do scalar components give the length of a vector?
\(\vec a=a_1\hat i+a_2\hat j+a_3\hat k,\quad |\vec a|=\sqrt{a_1^2+a_2^2+a_3^2}\)
Keep direction, change length
How do you turn a nonzero vector into a unit vector in the same direction?
\(\hat a=\frac{\vec a}{|\vec a|}\)
Direction cosines
How are direction cosines related to components?
\(l=\frac{a_1}{|\vec a|},\quad m=\frac{a_2}{|\vec a|},\quad n=\frac{a_3}{|\vec a|};\quad l^2+m^2+n^2=1\)
From P to Q
Which position vector do you subtract when the direction is P to Q?
\(\overrightarrow{PQ}=\vec q-\vec p\)
Head to tail
What is the triangle law, and how do you subtract vectors?
\(\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC},\quad \vec a-\vec b=\vec a+(-\vec b)\)
Internal section
R lies between P and Q, with PR:RQ=m:n. What is its position vector?
\(\vec r=\frac{n\vec p+m\vec q}{m+n}\)
External section and midpoint
What changes for external division? What is the midpoint formula?
\(\vec r=\frac{m\vec q-n\vec p}{m-n},\qquad \vec r_{mid}=\frac{\vec p+\vec q}{2}\)
Dot product
What does the dot product measure, and how is it calculated?
\(\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta=a_1b_1+a_2b_2+a_3b_3\)
Angle and perpendicularity
How do you recover the angle? When does dot product zero mean perpendicular?
\(\theta=\cos^{-1}\!\left(\frac{\vec a\cdot\vec b}{|\vec a||\vec b|}\right),\quad \vec a\perp\vec b\iff\vec a\cdot\vec b=0\)
Projection onto a direction
For the scalar projection of a onto b, whose magnitude is in the denominator?
\(\vec a\cdot\hat b=\frac{\vec a\cdot\vec b}{|\vec b|}\)
🎯 CBSE 2026 Board PYQs from Vector Algebra
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q13
The value of p for which vectors î + 2ĵ + 3k̂ and 2î − p ĵ + k̂ are perpendicular to each other is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q14
The value of m for which the points with position vectors −î − ĵ + 2k̂, 2î + m ĵ + 5k̂ and 3î + 11ĵ +…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q15
If |a| = 8, |b| = 3 and |a × b| = 12, then the value of |a · b| is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q23
Find the vector of magnitude 14 in the direction of QP, where P and Q are the points (1, 3, 2) and (−1…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q24
Vectors a = 3î − 2ĵ + 2k̂ and b = î + 2k̂ represent the two adjacent sides of a parallelogram. Find the…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q6
The value of p for which vectors î + 2ĵ + 3k̂ and 2î − p ĵ + k̂ are perpendicular to each other is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q7
The value of m for which the points with position vectors −î − ĵ + 2k̂, 2î + m ĵ + 5k̂ and 3î + 11ĵ +…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q8
If |a| = 8, |b| = 3 and |a × b| = 12, then the value of |a · b| is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q21
A vector of magnitude 14 has direction ratios (2,3,−6). Find its scalar projection on the positive…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q22
Vectors a = 3î − 2ĵ + 2k̂ and b = î + 2k̂ represent the two adjacent sides of a parallelogram. Find the…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q1
If |a| = 8, |b| = 3 and |a × b| = 12, then the value of |a · b| is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q17
The value of p for which vectors î + 2ĵ + 3k̂ and 2î − p ĵ + k̂ are perpendicular to each other is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q18
The value of m for which the points with position vectors −î − ĵ + 2k̂, 2î + m ĵ + 5k̂ and 3î + 11ĵ +…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q24
Using vectors, find the area of the triangle with A=(1,2,3), B=(2,−1,4), and C=(4,5,−1).
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q25
Vectors a = 3î − 2ĵ + 2k̂ and b = î + 2k̂ represent the two adjacent sides of a parallelogram. Find the…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q13
If |a|=5 and −2≤λ≤1, find the sum of the largest and smallest values of |λa|.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q14
Which vector of magnitude 3 makes equal angles with the x- and y-axes and is perpendicular to the z-axis?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q19
Assertion: |a×b|²+(a·b)²=|a|²|b|². Reason: |a×b|=(a·b) tan θ for θ≠π/2, where θ is the angle between…
📝 Chapterwise Practice Questions — Vector Algebra (124 Questions)
Class 12 Vector Algebra · Mcq Single Correct
Find |a × b|² + (a · b)².
Class 12 Vector Algebra · Mcq Single Correct
Find the area of triangle OAB, where OA = (1,2,3) and OB = (−3,−2,1).
Class 12 Vector Algebra · Mcq Single Correct
For nonzero vectors a,b with angle θ ∈ [0,π], when is a · b ≥ 0?
Class 12 Vector Algebra · Mcq Single Correct
For a vector a, find (a · i)i + (a · j)j + (a · k)k.
Class 12 Vector Algebra · Mcq Single Correct
Which vector has magnitude 9 and the same direction as (1,−2,2)?
Class 12 Vector Algebra · Mcq Single Correct
Find the position vector dividing the segment from 2a − 3b to a + b internally in the ratio 3:1.
Class 12 Vector Algebra · Mcq Single Correct
Find the direction cosines of BA when A = (1,2,−1), B = (3,4,0).
Class 12 Vector Algebra · Mcq Single Correct
Given |a| = √3, |b| = 4 and a · b = 2√3, find their angle.
Class 12 Vector Algebra · Mcq Single Correct
Find p if (2,p,1) and (−4,−6,26) are perpendicular.
Class 12 Vector Algebra · Mcq Single Correct
Find λ if (3,−6,1) and (2,−4,λ) are parallel.
Class 12 Vector Algebra · Mcq Single Correct
Find the area of triangle OAB with OA = (2,−3,2), OB = (2,3,1).
Class 12 Vector Algebra · Mcq Single Correct
For any vector a, find |a × i|² + |a × j|² + |a × k|².
Class 12 Vector Algebra · Mcq Single Correct
Given |a| = 10, |b| = 2 and a · b = 12, find |a × b|.
Class 12 Vector Algebra · Mcq Single Correct
Find the scalar projection of (1,−2,1) onto (4,−4,7).
Class 12 Vector Algebra · Mcq Single Correct
Unit vectors a,b,c satisfy a + b + c = 0. Find a · b + b · c + c · a.
Class 12 Vector Algebra · Mcq Single Correct
For b ≠ 0, which expression is the vector projection of a onto b? Here b̂ = b/|b|.
Class 12 Vector Algebra · Mcq Single Correct
If |a| = 4 and −3 ≤ λ ≤ 2, find the range of |λa|.
Class 12 Vector Algebra · Mcq Single Correct
How many unit vectors are perpendicular to both (2,1,2) and (0,1,1)?
Class 12 Vector Algebra · Mcq Single Correct
Find the point dividing the segment from a + b to 2a − b internally in the ratio 1:2.
Class 12 Vector Algebra · Mcq Single Correct
Nonzero vectors a,b have zero scalar projection of a onto b. Find their angle.
Class 12 Vector Algebra · Mcq Single Correct
In triangle ABC, AB = (1,1,2), AC = (3,−1,4). If D is the midpoint of BC, find AD.
Class 12 Vector Algebra · Mcq Single Correct
Let a = (a₁,a₂,a₃), b = (b₁,b₂,b₃), with b₁b₂b₃ ≠ 0. Which condition is necessary and sufficient for a…
Class 12 Vector Algebra · Mcq Single Correct
Find the magnitude of (6,−2,3).
Class 12 Vector Algebra · Mcq Single Correct
Find the unit vector along PQ, P = (2,1,−1), Q = (4,4,−7).