Question

You are given a vector, a⃗ =(156)i , and an unknown vector, b⃗ , that is perpendicular to a⃗ . The cross-product of these two vectors is a⃗ ×b⃗ =(98.7)k . What is the y component of the vector b⃗ ?

Answer 1

The y component of vector b⃗, given that it is perpendicular to a⃗ and that the cross-product is (98.7)k, is 0.633.

Step-by-step explanation:

To determine the y component of vector b⃗, given that vector a⃗ is (156)i and a⃗ × b⃗ is (98.7)k, we use the properties of the cross product. Let's assume vector b⃗ has components Bxî + Byĵ + Bzâk. Since a⃗ and b⃗ are perpendicular, the cross product a⃗ × b⃗ would equal the magnitude of a⃗ times the magnitude of b⃗ times the sine of 90 degrees (which is 1) in the k-direction.

Given that the cross product results in a k-vector, the x and z components of vector b⃗ must be zero. Therefore, vector b⃗ could be expressed as 0î + Byĵ + 0âk. The magnitude of the vector product, 98.7, is equal to the product of the magnitude of a⃗ (which is 156) and the magnitude of b⃗ along y (which we denote as By).

The cross product in terms of components is given by:

• a⃗ = 156î + 0ĵ + 0âk
• b⃗ = 0î + Byĵ + 0âk
• a⃗ × b⃗ = (156 * 0 - 0 * 0) î + (0 * 0 - 0 * 156) ĵ + (156 * By - 0 * 0) âk = (156 * By) âk

Setting the magnitude of the vector product equal to 98.7, we have 156 * By = 98.7. Therefore, the y component of vector b⃗, By, is 98.7 / 156, or approximately 0.633.