Parallelogram Law of Vector Addition

This law is just another way of understanding vector addition. This law states that if two vectors acting on the same point are represented by the sides of the parallelogram, then the resultant vector of these vectors is represented by the diagonals of the parallelograms. The figure below shows these two vectors represented on the side of the parallelogram. 

Parallelogram Law of Vector Addition

Also, Check:

Examples on Scalar and Vector

Example 1: Find the magnitude of v = i + 4j. 

Solution: 

|v| = �2+�2a2+b2​

a = 1, b = 4

|v| = 12+4212+42​

|v| = 12+4212+42​

|v| = √17

Example 2: A vector is given by, v = i + 4j. Find the magnitude of the vector when it is scaled by a constant of 5. 

Solution: 

|v| = �2+�2a2+b2​

5|v| = |5v| 

a = 1, b = 4

|5v|

|5(i + 4j)| 

|5i + 20j| 

|v| = 52+20252+202​

|v| = 25+40025+400​

|v| = √425

Example 3: A vector is given by, v = i + j. Find the magnitude of the vector when it is scaled by a constant of 0.5. 

Solution: 

|v| = �2+�2a2+b2​

0.5|v| = |0.5v| 

a = 1, b = 1

|0.5v|

|0.5(i + j)| 

|0.5i + 0.5j| 

|v| = 0.52+0.520.52+0.52​

|v| = 0.25+0.250.25+0.25​

|v| = √0.5

Example 4: Two vectors with magnitude 3 and 4. These vectors have a 90° angle between them. Find the magnitude of the resultant vectors. 

Solution: 

Let the two vectors be given by p and q. Then resultant vector “r” is given by, 

∣�∣=∣�∣2+∣�∣2+2∣�∣∣�∣���(�)∣r∣=∣p∣2+∣q∣2+2∣p∣∣qcos(θ)​

|p| = 3, |q| = 4 and �=90�θ=90o

∣�∣=∣�∣2+∣�∣2+2∣�∣∣�∣���(�)∣r∣=∣p∣2+∣q∣2+2∣p∣∣qcos(θ)​

∣�∣=∣3∣2+∣4∣2+2∣3∣∣4∣���(90)∣r∣=∣3∣2+∣4∣2+2∣3∣∣4∣cos(90)​

∣�∣=∣3∣2+∣4∣2∣r∣=∣3∣2+∣4∣2​

∣�∣=9+16∣r∣=9+16​

∣�∣=9+16                   ∣r∣=9+16​  

|r| = 5

Example 5: Two vectors with magnitude 10 and 9. These vectors have a 60° angle between them. Find the magnitude of the resultant vectors. 

Solution: 

Let the two vectors be given by p and q. Then resultant vector “r” is given by, 

∣�∣=∣�∣2+∣�∣2+2∣�∣∣�∣���(�)∣r∣=∣p∣2+∣q∣2+2∣p∣∣qcos(θ)​

|p| = 10, |q| = 9 and �=60�θ=60o

∣�∣=∣�∣2+∣�∣2+2∣�∣∣�∣���(�)∣r∣=∣p∣2+∣q∣2+2∣p∣∣qcos(θ)​

∣�∣=∣10∣2+∣9∣2+2∣10∣∣9∣���(60)∣r∣=∣10∣2+∣9∣2+2∣10∣∣9∣cos(60)​

∣�∣=∣10∣2+∣9∣2+(10)(9)∣r∣=∣10∣2+∣9∣2+(10)(9)​

∣�∣=100+81+90∣r∣=100+81+90​

∣�∣=271  ∣r∣=271​  


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