a = i + 2 j - 2 k and b = 2 i - 2 j - k. Which of the following statements are true? I. a · b = 0. II. The projection of a on b is 3. III. | a - b | = 3√(2). IV. a + b and a - b are perpendicular.

Question 82

MediumOriginal question

$\vec{a} = \hat{i} + 2\hat{j} - 2\hat{k}$ and $\vec{b} = 2\hat{i} - 2\hat{j} - \hat{k}$ . Which of the following statements are true?
I. $\vec{a} \cdot \vec{b} = 0$ .
II. The projection of $\vec{a}$ on $\vec{b}$ is 3.
III. $\lvert \vec{a} - \vec{b} \rvert = 3\sqrt{2}$ .
IV. $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ are perpendicular.

I and III onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerI, III and IV onlyCorrect answerYour answerIII and IV onlyCorrect answerYour answer

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CorrectNot quiteAnswer C

Correct answer: C

I: $2 - 4 + 2 = 0$ , true; so the projection of $\vec{a}$ on $\vec{b}$ is $\frac{0}{3} = 0$ , and II is false.
III: $\lvert \vec{a} - \vec{b} \rvert^{2} = 9 - 0 + 9 = 18$ , true. IV: $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = \lvert \vec{a} \rvert^{2} - \lvert \vec{b} \rvert^{2} = 9 - 9 = 0$ , true.

Worth noting

IV holds because the magnitudes are equal, not because $\vec{a}$ and $\vec{b}$ are perpendicular.

Concept: What a scalar product tells
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe scalar product

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