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Let a and b be positive real numbers. PQ = ai + bj and PS = ai - bj represent adjacent sides of a parallelogram PQRS. The vectors →u and →v are the projections of →w = i + j onto PQ and PS, respectively. If the condition |→u| + |→v| = |→w| is satisfied and the area of parallelogram PQRS equals 8, which of the following statements is/are correct?

1. a + b equals 4
2. a minus b equals 2
3. The diagonal PR of parallelogram PQRS has a length of 4
4. →w bisects the angle between PQ and PS

Correct answer: a + b equals 4

Solution

The condition |→u| + |→v| = |→w| implies a specific geometric relationship between the vectors. Given the area of the parallelogram is 8, solving for the side lengths shows that a + b equals 4.

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