Preloader
  • By Admin Koushi
  • (0) comments
  • March 29, 2025

Vector dot product

Vector dot product or scalar product:

The dot or scalar product of two vectors and  is defined as the product of the magnitudes of and  and the cosine of the angle between them.

If A and B creates angle θ then, . = AB .

Properties of dot product:

(i) . = BA  = AB  = . [dot product is commutative]

(ii) .(+) = . + . [dot product is distributive]

(iii) If the angle between and  is  then, .  = ABcos = AB 

(iv) . = AAcos =

(v) If the angle between and is then, . = ABcos  = 0

(vi) . = . = . = 1.1cos = 1

(vii) . = . = . = 1.1cos = 0

Vector dot product or scalar product is used to calculate work done by a force, projection of a vector, angle between two vectors. We can show two vectors are mutually perpendicular using dot product of vector.

Concepts on dot product:

1. Work done: A force = (2î + 3ĵ – k̂) N is applied on a body to move it from point P (1, 2, -1) m to point Q (2, 4, 1) m. What is the work done required?

Force  = (2î + 3ĵ – k̂) N and displacement of the body = (2-1 ) î + (4 – 2)ĵ + (1 + 1)k̂ = (î + 2ĵ + 2k̂) m.

Work done W = . = (2î + 3ĵ – k̂).(î + 2ĵ + 2k̂) = 2 + 6 – 2 = 6J.

2. Projection: Projection of on or component of  on . PQ represents the magnitude of . PR and QS are two perpendiculars drawn on  and PT is parallel to RS. The component of  on is RS = PT = PQcos  [where  is the angle between  and ] .

= A.1.cos = .

Example: What is the component of  = 2î + 3ĵ + k̂ on  = î + 2ĵ + 2k̂?

Component of  on  = . =  =  = .

Vector component of  on  is (. ) = () = .

3. Angle between two vectors: If is the angle between and  then  = .

Example: What is the angle between  = 2î + 3ĵ + k̂ and  = î + 2ĵ – 2k̂?

 =  =  = .

4. Perpendicular vectors: If and are perpendicular to each other then, .  = 0.

Example: What is the value of n so that  = 2î – 3ĵ + k̂ and  = î + nĵ + 2k̂ are perpendicular to each other?

Here (2î – 3ĵ + k̂).( î + nĵ + 2k̂) = 0

Or, 2 – 3n +2 = 0  

n = .

5. Equation of a plane and distance of plane from origin:

 = 2î+3ĵ+2k̂ is perpendicular to a plane. The terminal point of another vector = î+2ĵ+k̂ touches the plane. Find the equation of the plane.

Let us consider, Q is a point on the plane with coordinates (x, y, z). The position vector of point Q is xî + yĵ + zk̂.

 = (1 – x)î + (2 – y)ĵ+(1 – z)k̂ is lying on the plane which is perpendicular to .

Therefore, . = 0

Or, (2î + 3ĵ + 2k̂).[(1 – x)î + (2 – y)ĵ + (1 – z)k̂] = 0

Or, 2(1 – x) + 3(2 – y) + 2(1 – z) = 0

Or, 2 – 2x + 6 – 3y + 2 – 2z = 0

 2x + 3y + 2z = 10

The distance of plane from origin is projection of  on  = . =  = .

Admin Koushi

previous post next post

Leave a comment

Your email address will not be published. Required fields are marked *

contact info

subscribe newsletter

Subscribe to get our Latest Updates

Get updates On New Courses and News

© 2018 – 2025 Koushi All Rights Reserved