What are the values of for which the angle between the vectors and is obtuse?

. For the angle between two vectors to be obtuse, their dot product must be strictly negative. . This gives . The roots are and . The inequality holds for .

What are the values of x for which the angle between the vectors 2x^{2}\hat{i}+3x\hat{j}+\hat{k} and \hat{i}-2\hat{j}+x^{2}\hat{k} is obtuse?

A. 0 \lt x \lt 2 ✓
B. x \lt 0
C. x \gt 2
D. 0 \leq x \leq 2

Correct Answer: A. 0 \lt x \lt 2

Explanation

For the angle between two vectors to be obtuse, their dot product must be strictly negative. (2x^2)(1) + (3x)(-2) + (1)(x^2) \lt 0. This gives 2x^2 - 6x + x^2 \lt 0 \implies 3x^2 - 6x \lt 0 \implies 3x(x - 2) \lt 0. The roots are x=0 and x=2. The inequality holds for 0 \lt x \lt 2.

What are the values of x for which the angle between the vectors 2x^{2}\hat{i}+3x\hat{j}+\hat{k} and \hat{i}-2\hat{j}+x^{2}\hat{k} is obtuse?

Explanation

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