Chapter 10 Vector Algebra

Notes

Vector Algebra One Shot Notes

Scalar and Vector

Scalar: Jisme only magnitude hota hai. Example: mass, time, temperature, speed

Vector: Jisme magnitude aur direction dono hote hain. Example: displacement, velocity, force, acceleration

Vector representation

A vector in 3D: a = xi + yj + zk

Here: i, j, k are unit vectors along x-axis, y-axis, z-axis.

Magnitude of vector

If a = xi + yj + zk

|a| = √(x2 + y2 + z2)

Example: a = 2i + 3j + 6k

|a| = √(4 + 9 + 36) = 7

Unit vector

Unit vector means magnitude 1.

Unit vector in direction of a: a cap = a / |a|

Example: a = 3i + 4j

|a| = 5

Unit vector = 3/5 i + 4/5 j

Equal vectors

Two vectors are equal if: Their magnitude same ho Their direction same ho

Negative vector

Negative vector ka magnitude same hota hai but direction opposite hoti hai.

If a = 2i + 3j

Then -a = -2i - 3j

Addition of vectors

If a = x1i + y1j + z1k b = x2i + y2j + z2k

a + b = (x1 + x2)i + (y1 + y2)j + (z1 + z2)k

Subtraction of vectors

a - b = (x1 - x2)i + (y1 - y2)j + (z1 - z2)k

Multiplication by scalar

If a = xi + yj + zk

ka = kxi + kyj + kzk

Position vector

Point P(x, y, z) ka position vector: OP = xi + yj + zk

Vector joining two points

If A(x1, y1, z1) and B(x2, y2, z2)

AB = (x2 - x1)i + (y2 - y1)j + (z2 - z1)k

Direction cosines

If a vector makes angles α, β, γ with x-axis, y-axis, z-axis, then:

l = cos α m = cos β n = cos γ

These are called direction cosines.

For vector a = xi + yj + zk:

l = x/|a| m = y/|a| n = z/|a|

Important property: l2 + m2 + n2 = 1

Direction ratios

For vector a = xi + yj + zk, direction ratios are: x, y, z

Direction ratios proportional hote hain.

Dot product or scalar product

If angle between a and b is θ:

a . b = |a||b| cos θ

If a = x1i + y1j + z1k b = x2i + y2j + z2k