Assertion (A): If a × b = 0 and a · b = 0, then a = 0 or b = 0. Reason (R): For non-zero vectors, a × b = 0 means they are parallel, and a · b = 0 means they are perpendicular.

Question 107

MediumOriginal question

Assertion (A): If $\vec{a} \times \vec{b} = \vec{0}$ and $\vec{a} \cdot \vec{b} = 0$ , then $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0}$ .
Reason (R): For non-zero vectors, $\vec{a} \times \vec{b} = \vec{0}$ means they are parallel, and $\vec{a} \cdot \vec{b} = 0$ means they are perpendicular.

Both A and R are true, and R is the correct explanation of A.Correct answerYour answerBoth A and R are true, but R is not the correct explanation of A.Correct answerYour answerA is true, but R is false.Correct answerYour answerA is false, but R is true.Correct answerYour answer

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CorrectNot quiteAnswer A

Correct answer: A

R is true: $\lvert \vec{a} \rvert \lvert \vec{b} \rvert \sin \theta = 0$ forces $\sin \theta = 0$ , and $\lvert \vec{a} \rvert \lvert \vec{b} \rvert \cos \theta = 0$ forces $\cos \theta = 0$ .
No angle has both sine and cosine 0, so two non-zero vectors cannot do both: one of them is $\vec{0}$ . A is true, and R explains it.

Concept: When both products vanish
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsThe vector product

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