Question

⃗i and ⃗j are unit vectors along x- and y-axis respectively. What is the magnitude and direction of the vectors  ⃗⃗i+ ⃗j and ⃗i − ⃗j ? What are the components of a vector ⃗A = 2⃗ and 3⃗ along the directions of  ⃗⃗i + ⃗j and ⃗i – ⃗j? [You may use graphical method]

Answer 1

Consider a vector ⃗p, given as:

On comparing the components on both sides, we get:

Hence, the magnitude of the vector ⃗⃗i+ ⃗j is √2 .

Let be the angle made by the vector ⃗p, with the x-axis, as shown in the following figure.

Hence, the magnitude of the vector ⃗⃗i − ⃗j is √2.

Let be the angle made by the vector ⃗Q, with the x- axis, as shown in the following figure.

On comparing the coefficients of  ⃗i and  ⃗j, we have: 

Let _⃗  make an angle with the x-axis, as shown in the following figure.

Angle between the vectors

 

Component of vector  ⃗A, along the direction of  ⃗p, making and angle .

Let θ be the angle between the vectors 

Component of vector  ⃗A, along the direction of Q ⃗, making and angle .