Question

7. The vectors d &e are [- 4, 1] \&[1,4] respectively. Find the angle b/n d&e( use dot product)

Answer 1

90° or π/2 rad

Step-by-step explanation:

Given two vectors:

`\displaystyle{ \vec d = - 4 \hat i + \hat j} \ \: \text{and} \ \: \displaystyle{ \vec e = \hat i + 4 \hat j}`

We can find the angle by applying dot product:

`\displaystyle \cos \theta`

Find magnitude of two vectors:

`\displaystyle{ | \vec d| = \sqrt{ {( - 4)}^(2) + {1}^(2) }} \\ \\ \displaystyle \vec d`

And

`\displaystyle{ | \vec e| = \sqrt{ 1^(2) + {4}^(2) }} \\ \\ \displaystyle \vec e`

Thus, we now have:

`\displaystyle{ \vec d \cdot \vec e = √(17) \cdot √(17) \cos \theta} \\ \\ \displaystyle{ \vec d \cdot \vec e = 17 \cos \theta}`

Product of two vectors are:

`\displaystyle{ \vec d \cdot \vec e = d_xe_x + d_ye_y}`

Therefore:

`\displaystyle{( - 4)(1) + (1)(4) = 17 \cos \theta} \\ \\ \displaystyle{ - 4 +4= 17 \cos \theta} \\ \\ \displaystyle{0= 17 \cos \theta} \\ \\ \displaystyle{0 =\cos \theta}`

Therefore, the angle would be 90° or π/2 since we know that cos90° = 0.