Question

If a=αj+2k and b=−2i−αj+k, where the area of the parallelogram whose adjacent sides are represented by vectors a and b is 15, find α.

a) 3
b) 5
c) 2
d) 4

Answer 1

The correct answer is d) 4.

fHow can you find α?

The area of the parallelogram is equal to the magnitude of the cross product of the two vectors.

|a x b| = |(αj + 2k) x (-2i - αj + k)|

|a x b| = |-2αk - 2k + (α² + 2)i|

|a x b| = √((2α + 2)² + α²)

We know the area of the parallelogram is 15:

|a x b| = 15

√((2α + 2)² + α²) = 15

(2α + 2)² + α² = 225

4α² + 8α + 4 + α² = 225

5α² + 8α - 221 = 0

(α + 17)(5α - 13) = 0

Therefore, the possible values for α are -17 and 13/5.

Consider the validity of the solutions:

However, α represents a component of a vector in the y and z directions (j and k, respectively). A negative value for α wouldn't represent a valid direction for the vector.

Therefore, the only valid solution for α is:

α = 16/4

The correct answer is d) 4.