Question

Consider the vectors Q 1 =( 2 , 2 ,0) and Q 2 =( 2 +2, 2 −2,2). please answer Qart b) (b) (1/2 pt) Use the inner product to determine the angle between Q 1 and Q 2 and use the cross product to determine the angle between the two vectors.

Answer 1

The angle between Q1 and Q2 using the inner product is approximately 39.2°. The angle between Q1 and Q2 using the cross product is 0°.

Step-by-step explanation:

To find the angle between two vectors, Q1 and Q2, we can use the inner product. The inner product of two vectors A and B is given by A · B = |A| |B| cosθ, where θ is the angle between them.
For Q1 =(2, 2, 0) and Q2 =(2 + 2, 2 - 2, 2), we can calculate the inner product as follows:

Q1 · Q2 = (2)(4) + (2)(0) + (0)(2) = 8
|Q1| = √(2^2 + 2^2 + 0^2) = √8
|Q2| = √(4^2 + (-2)^2 + 2^2) = 6
Now, we can use the formula A · B = |A| |B| cosθ to find the angle:

8 = √8 * 6 * cosθ
cosθ = 8 / (√8 * 6) = √2 / 3
θ = cos^(-1)(√2 / 3) ≈ 39.2°

To find the angle between two vectors using the cross product, we can use the formula A × B = |A| |B| sinθ.
For Q1 and Q2, we can calculate the cross product as follows:

Q1 × Q2 = (2)(2 - 2) - (2)(2 + 2) + (2)(2 - 2) = 0
|Q1| = √8
|Q2| = 6
Now, we can use the formula A × B = |A| |B| sinθ to find the angle:

0 = √8 * 6 * sinθ
sinθ = 0
θ = sin^(-1)(0) = 0°