IIT JEE Main Maths -Unit 12- Scalars and vectors- Study Notes-New Syllabus
IIT JEE Main Maths -Unit 12- Scalars and vectors – Study Notes – New syllabus
IIT JEE Main Maths -Unit 12- Scalars and vectors – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Scalars and vectors
Vector Algebra: Scalars and Vectors
Vector Algebra is one of the most scoring chapters in JEE. It deals with physical and geometrical quantities which have magnitude alone or both magnitude and direction.
Scalars
A scalar quantity is one that has only magnitude and no direction.
Common Scalar Quantities:
- Mass
- Time
- Temperature
- Speed
- Distance
- Density
Properties of Scalars:
- Fully described by a real number.
- Added, subtracted, multiplied using normal algebraic rules.
- Represented by plain symbols such as \( a, m, t, k \).
Vectors
A vector quantity has both magnitude and direction.
Common Vector Quantities:
- Displacement
- Velocity
- Acceleration
- Force
- Momentum
Representation:
- Vector \( \vec{a} \) is written as \( \vec{a} = \langle a_1, a_2, a_3 \rangle \).
- Geometrically represented by a directed line segment.
- Magnitude (length) of vector \( \vec{a} = (a_1, a_2, a_3) \) is \( |\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \).
Types of Vectors
- Zero vector: \( \vec{0} = (0, 0, 0) \)
- Unit vector: A vector with magnitude 1 \( \hat{a} = \dfrac{\vec{a}}{|\vec{a}|} \)
- Collinear vectors: Parallel or anti parallel vectors
- Equal vectors: Same magnitude and direction
- Negative vectors: Same magnitude, opposite direction
Position Vector
The position vector of a point \( P(x, y, z) \) is
\( \vec{OP} = (x, y, z) \)
Direction Angles and Direction Cosines
If a vector makes angles \( \alpha, \beta, \gamma \) with the positive \( x, y, z \) axes respectively, then
\( \cos\alpha = \dfrac{a_1}{|\vec{a}|},\quad \cos\beta = \dfrac{a_2}{|\vec{a}|},\quad \cos\gamma = \dfrac{a_3}{|\vec{a}|} \)
And
\( \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \)
Examples
Example
A vector is given by \( \vec{a} = (3, 4, 0) \). Find its magnitude.
▶️ Answer / Explanation
\( |\vec{a}| = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{25} = 5 \)
Example
Find the unit vector in the direction of \( \vec{a} = (2, -1, 2) \).
▶️ Answer / Explanation
\( |\vec{a}| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{9} = 3 \)
Unit vector \( \hat{a} = \dfrac{1}{3}(2, -1, 2) = \left( \dfrac{2}{3}, -\dfrac{1}{3}, \dfrac{2}{3} \right) \)
Example
A vector has direction cosines \( \left( \dfrac{2}{3}, \dfrac{1}{3}, \dfrac{2}{3} \right) \). What is the vector of magnitude 6 with these direction cosines?
▶️ Answer / Explanation
If \( \hat{a} = (l, m, n) \), and magnitude is \( k \), then
\( \vec{a} = k(l, m, n) \)
So,
\( \vec{a} = 6\left( \dfrac{2}{3}, \dfrac{1}{3}, \dfrac{2}{3} \right) = (4, 2, 4) \)
Required vector = \( (4, 2, 4) \)
Notes and Study Materials
- Concepts of Vector Algebra
- Vector Algebra Master File
- Addition and scalar multiplication Notes
- Multiplication of Vector Notes
- Vector Algebra Formulae
- Vector Algebra Reference Book
- Vector Algebra Past Many Years Questions and Answer
Examples and Exercise
- Practice Paper 1
- Practice Paper 2
- Practice Paper 3
- Practice Paper 4
- Practice Paper 5
- Practice Paper 6
- Practice Paper 7
- Practice Paper 8
IIT JEE (Main) Mathematics ,” Vector Algebra” Notes ,Test Papers, Sample Papers, Past Years Papers , NCERT , S. L. Loney and Hall & Knight Solutions and Help from Ex- IITian
About this unit
Vectors and scalars, the addition of vectors.Components of a vector in two dimensions and three-dimensional space.Scalar and vector products, scalar and vector triple product.
IITian Academy Notes for IIT JEE (Main) Mathematics – Vector Algebra
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IIT JEE (Main) Mathematics, Vector Algebra Solved Examples and Practice Papers.
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S. L. Loney IIT JEE (Main) Mathematics
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Hall & Knight IIT JEE (Main) Mathematics
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IIT JEE (Main) Mathematics Assignments
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Past Many Years (40 Years) Questions IIT JEE (Main) Mathematics Solutions Vector Algebra
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