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= 2 i + 3 j along the directions of i + j + and i - j ? [You may use graphical method]

Answer 1

Consider a vector 𝑃⃗, given as:
P = i+j
pxi + pyj = i+j
On comparing the components on both sides, we get:
Px = Py =1
|P| = $\sqrt{P_x^2+P_y^2}$= $\sqrt{(1)^2+(1)^2}$=$\sqrt{2}$ ...(i)

Hence, the magnitude of the vector 𝑖 ⃗ + 𝑗⃗ is $\sqrt{2}$ .

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

∴tanθ = $\frac{P_y}{P_x}$= 45 ...(ii)

Hence, the vector i + j makes an angle of 45 with the x- axis.

Let Q = i - j
Qx, i - Q , j = i-j
Qx = Qy = 1
|Q| = $\sqrt{ Q_x^2 + Q_y^2}$ = $\sqrt{2}$ ....(iii)

Hence, the magnitude of the vector 𝑖 ⃗⃗− 𝑗⃗ is $\sqrt{2}$.

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

∴tanθ = ($\frac{Q_y}{Q_x}$)
θ = - tan-1 ($\frac{-1}{-1}$) = -45 ...(iv)

Hence, the vector i - j makes an angle of 45 with the x- axis.

It is given that:
A = 2i + 3j
ax i + Ay j = 2i + 3j

On comparing the coefficients of 𝑖⃗ and 𝑗⃗, we have:
Ax = 2 and Ay = 3
|A| = √ 22 + 33 = √13

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

∴tanθ = (Ay/Ax)
θ = - tan-1 ($\frac{3}{2}$) = tan-1 (1.5) = 56.31

Angle between the vectors (2i + 3j ) and (i + j). θ= 56.31 -45 = 11.31
Component of vector 𝐴⃗, along the direction of 𝑃⃗⃗, making and angle 𝜃.
= (Acosθ) P = (Acos 11.31) $\frac{(i+j)}{\sqrt2}$.
= $\frac{\sqrt{13 \times 0.9806}}{ \sqrt{2 (i+j)}}$
= 2.5 (i+j)
=$\frac{25}{10}$ x $\sqrt{2}$
=$\frac{5}{\sqrt2}$ ...(v)

Let θ be the angle between the vectors (2i + 3j ) and (i-j)
θ = 45 + 56.31 = 101.31

Component of vector 𝐴⃗, along the direction of 𝑄⃗, making and angle 𝜃.

= (Acosθ) Q = (Acosθ) $\frac{i-j}{\sqrt2}$
= $\sqrt{13}$ cos(901.31) $\frac{(i-j)}{\sqrt2}$
=$\sqrt{\frac{13}{2}}$ sin 11.30 (i-j)
= -0.5 (i-j)
= $\frac{-5}{10}$ x $\sqrt{2}$
=$-\frac{1}{\sqrt2}$ ...(vi)