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]

Updated On: Oct 4, 2024
Hide Solution
collegedunia
Verified By Collegedunia

Solution and Explanation

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 βƒ— + 𝑗⃗ is \(\sqrt{2}\) .

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

∴tanΞΈ = \(\frac{P_y}{P_x}\)= 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 βƒ—βƒ—βˆ’ 𝑗⃗ is \(\sqrt{2}\)

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

∴tanθ = (\(\frac{Q_y}{Q_x}\))
ΞΈ = - 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| = βˆš 22 + 33 = βˆš13

Let 𝐴π‘₯ βƒ— make an angle πœƒ with the x-axis, as shown in the following figure.

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

Angle between the vectors (2i + 3j ) and (i + j). ΞΈ= 56.31 -45 = 11.31
Component of vector 𝐴⃗, along the direction of 𝑃⃗⃗, making and angle πœƒ. 
= (AcosΞΈ) 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 ΞΈ be the angle between the vectors (2i + 3j ) and (i-j)
ΞΈ = 45 + 56.31 = 101.31

Component of vector 𝐴⃗, along the direction of 𝑄⃗, making and angle πœƒ.

= (AcosΞΈ) Q = (AcosΞΈ) \(\frac{i-j}{\sqrt2}\)
\(\sqrt{13}\) 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)

Was this answer helpful?
1
1

Concepts Used:

Addition of Vectors

A physical quantity, represented both in magnitude and direction can be called a vector.

For the supplemental purposes of these vectors, there are two laws that are as follows;

  • Triangle law of vector addition
  • Parallelogram law of vector addition

Properties of Vector Addition:

  • Commutative in nature -

It means that if we have any two vectors a and b, then for them

\(\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{b}+\overrightarrow{a}\)

  • Associative in nature -

It means that if we have any three vectors namely a, b and c.

\((\overrightarrow{a}+\overrightarrow{b})+\overrightarrow{c}=\overrightarrow{a}+(\overrightarrow{b}+\overrightarrow{c})\)

  • The Additive identity is another name for a zero vector in vector addition.

Read More: Addition of Vectors