Angle between vectors...?

Question

I have slightly confused with this Physics problem!.....i cannot figure out how to do it

Two vectors A and B have magnitude A=3.06 and B=3.09. Their vector product is A x B = -5.09 k + 1.91 i. What is the angle between vectors A and B?

Answer 1

Let vector C = vector A cross vector B,
C = A x B

The magnitude of vector C is equal to the magnitude of vector A multiplied by the magnitude of vector B multiplied by the sine of the angle (θ) between the two vectors,
|C| = |A| * |B| * Sin (θ)
We can re-write this equation to solve for the angle, θ, to be,
θ = arcsin [ |C| / (|A| * |B|) ]

We are given the magnitudes of vectors A and B. We are also given vector C (from which we can find its magnitude).
The magnitude of vector C (|C|) is the square root of the sum of the squares of the components of vector C.
|C| = sqrt ((C_x)^2 + (C_y)^2 + (C_z)^2)
Where C_x, C_y, and C_z refer to the x, y, and z components of the vector.

|A| = 3.06 (given)
|B| = 3.09 (given)
|C| = 5.44 (calculated as shown above)

We now have all the information we need to find the angle between vectors A and B.
θ = arcsin [ |C| / (|A| * |B|) ]
θ = arcsin [ 5.44 / (3.06 * 3.09) ]
θ = 35.12°

Answer 2

Calculate the magnitude of A cross B.

|| A X B || = √[1.91² + 0² + (-5.09)²] = 5.4365614

You can find the angle θ between two vectors with the cross product.

|| A X B || = || A || || B || sinθ

sinθ = || A X B || / (|| A || || B ||)

sinθ = 5.4365614 / (3.06*3.09) = .5749689

θ = arcsin(.5749689) = 35.097458°

Answer 3

Use the dot product. Assuming i, j, and ok are the muse instruments vectors i(a million, 0, 0), j(0, a million, 0), ok(0, 0, a million) u(c, d).v(f, g) = cf + dg u.v = |u| |v| cos a, the place a is the perspective between them. |u| = sqrt(2^2 + 5^2 + 0) = sqrt(4 + 25) = sqrt(29) |v| = sqrt(3^2 + (-2)^2 + (-5)^2) = sqrt(9+4+25) = sqrt(38) u.v = 2 x 3 + 5 x -2 + 0 x -5 = 6 - 10 = -4 cos a = u.v / |u| |v| cos a = -4 / sqrt(29)sqrt(38) = -4 / sqrt(a million 102) a ~= a million.6916 radians ~= ninety six.9 levels