Class 11 Mathematics Vectors Notes
Complete typed notes based on the supplied handwritten Vectors PDF, including unit vectors, displacement, vector addition, scalar product, projection, angle between vectors and solved Exercise 9.
1. Unit Vector
A vector whose magnitude is unity is called a unit vector.
2. Displacement Along an Axis or on a Line
Let O be the origin. If the position vectors of points P and Q are OP⃗ = x₁ and OQ⃗ = x₂, then:
In three dimensions:
OQ⃗ = (x₂, y₂, z₂)
PQ⃗ = (x₂-x₁, y₂-y₁, z₂-z₁)
3. Addition of Two Vectors
4. Triangle Law of Vectors
If vectors are represented by two sides of a triangle taken in order, then their sum is represented by the third side taken in the same order.
5. Midpoint Formula
If C is the midpoint of AB, then:
6. Magnitude and Direction Cosines of a Vector
For a vector with components x, y and z, the direction cosines are:
m = y/√(x²+y²+z²)
n = z/√(x²+y²+z²)
The notes prove:
Parallelogram Example
If ABCD is a parallelogram and O is the point of intersection of its diagonals, the notes use the midpoint property of both diagonals to obtain the corresponding vector relation among OA⃗, OB⃗, OC⃗ and OD⃗.
Collinearity Example
The source checks whether three points are collinear by comparing the direction vectors between them. If one vector is a scalar multiple of another, the three points are collinear.
7. Scalar Product of Vectors
If a⃗ = (a₁, a₂) and b⃗ = (b₁, b₂), then:
In terms of the angle θ between them:
Properties of Scalar Product
- Commutative: a⃗ · b⃗ = b⃗ · a⃗
- Distributive: a⃗ · (b⃗ + c⃗) = a⃗ · b⃗ + a⃗ · c⃗
- Self product: a⃗ · a⃗ = |a⃗|²
Useful Identities
8. Angle and Projection
Angle Between Two Vectors
Projection
Parallel Vectors
If two vectors are parallel, θ = 0°, so:
Perpendicular Vectors
If two vectors are perpendicular, θ = 90°, so:
9. Exercise 9
1. Find the scalar product
(a)
b⃗ = 2i⃗ – 5j⃗ + k⃗
a⃗·b⃗ = 2 – 10 + 3 = -5
(b)
b⃗ = 7i⃗ – 5j⃗ + 2k⃗
a⃗·b⃗ = 0 – 10 + 10 = 0
2. Evaluate Vector Expressions
The notes use identities such as:
and scalar multiplication to simplify the given expressions.
3. Find the Angle Between Pairs of Vectors
The source uses:
One worked example gives:
θ = cos-1(√(2/3))
Another example has dot product 0, giving θ = π/2.
4. Find p if Two Vectors are Perpendicular
Using a⃗·b⃗ = 0, the handwritten solution obtains:
5. Find the Projection
The notes first calculate the scalar product and magnitude, then use:
6. Find the Angles of a Triangle Using Vectors
The source forms AB⃗, BC⃗ and CA⃗ from the coordinates of the three vertices, then applies the scalar-product formula to determine each angle.
7. If |a⃗+b⃗|² = |a⃗-b⃗|², prove a⃗ ⟂ b⃗
4a⃗·b⃗ = 0
a⃗·b⃗ = 0
Therefore a⃗ is perpendicular to b⃗.
8. If (a⃗+b⃗)·(a⃗-b⃗)=0, prove |a⃗|=|b⃗|
|a⃗|² = |b⃗|²
|a⃗| = |b⃗|
9. Prove a Dot-Product Identity
The source proves this by expanding both squared magnitudes.
10. Parallelogram Law Using Vectors
If AC and BD are diagonals of a parallelogram ABCD, the notes derive vector expressions for the diagonals and prove identities relating their squared lengths to the sides.
10. Vector Method in a Triangle
11. Prove Relations in Triangle ABC by Vector Method
The source derives cosine-rule type identities including:
12. Prove Trigonometric Identities Using Vectors
Using position vectors and the scalar product, the notes prove:
and similarly:
Note: This typed version follows the uploaded 21-page Vectors PDF closely. The Google Drive PDF supplied in your message is embedded at the top.
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