Question

Obtain the angle between A+B and A-B if A= 2i + 3j and B= i - 2j.​

Answer 1

Given :

Vector A is given as :

A = 2i + 3j

vector B is given as :

B = i - 2j

To Find :

What is the angle between  $A+B$ and $A-B$ ?

Solution :

The vector A+B will be given by :

A+B = (2i + 3j) + (i -2j)

       = 3i +j

And vector A-B will be given by :

A-B = (2i+3j) - (i-2j)

      =i + 5j

We know  that cos of the angle between two vectors is given by the ratio of dot product of two vectors to the product of the magnitudes of the two vectors .

So, if the angle between A+B and A-B is $\theta$ the we have :

$cos\theta = \frac{(A+B).(A-B)}{|A|\times |B|}$

Or , $cos\theta = \frac{(3i+j).(i+5j)}{\sqrt{3^2+1^2}\times \sqrt{1^2+5^2}}$

Or, $cos\theta =\frac{3 + 5}{\sqrt{10}\times \sqrt{26}}$

Or, $cos\theta = \frac{15}{\sqrt{260}}$

So, $\theta = cos^{-1}\frac{15}{2\sqrt{65}}$

Hence the angle between A+B and A-B is $\theta = cos^{-1}\frac{15}{2\sqrt{65}}$

Answer 2

The angle between A + B and A - B is 0.57

Given:

A = 2i + 3j

B = i - 2j

Step-by-step explanation:

A + B = (2i + 3j) + (i - 2j) = 3i + j

A - B = (2i + 3j) - (i - 2j) = i + 5j

The angle between two vectors is given by the formula:

θ = cos⁻¹ (((A + B).(A - B))/(|A + B|.|A - B|))

Now,

(A + B).(A - B) = (3i + j).(i + 5j) = 3 + 5 = 8

|A + B| = √((3)² + (1)²) = √(9 + 1) = √(10) = 3.16

|A - B| = √((1)² + (5)²) = √(1 + 26) = √(27) = 3

On substituting the values, we get,

θ = cos⁻¹ ((8)/(3.16 × 3))

θ = cos⁻¹ ((8)/(9.48))

θ = cos⁻¹ (0.84)

∴ θ = 0.57