For a = -2 i + j + 2 k, which of the following statements are true? I. | a | = 3. II. The direction cosines of a are -2, 1, 2. III. a makes an obtuse angle with the x-axis. IV. 1/9(-2 i + j + 2 k) is the unit vector along a.

Question 25

MediumOriginal question

For $\vec{a} = -2\hat{i} + \hat{j} + 2\hat{k}$ , which of the following statements are true?
I. $\lvert \vec{a} \rvert = 3$ .
II. The direction cosines of $\vec{a}$ are $-2, 1, 2$ .
III. $\vec{a}$ makes an obtuse angle with the $x$ -axis.
IV. $\frac{1}{9}(-2\hat{i} + \hat{j} + 2\hat{k})$ is the unit vector along $\vec{a}$ .

I, II and III onlyCorrect answerYour answerI, III and IV onlyCorrect answerYour answerI and III onlyCorrect answerYour answerI and II onlyCorrect answerYour answer

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CorrectNot quiteAnswer C

Correct answer: C

$\lvert \vec{a} \rvert = \sqrt{4 + 1 + 4} = 3$ , so I is true; the direction cosines are $-\frac{2}{3}, \frac{1}{3}, \frac{2}{3}$ , and II lists the direction ratios.
III: $l = -\frac{2}{3} < 0$ , an obtuse angle, so it is true. IV divides by $9$ , the square of the magnitude, so it is false.

Worth noting

Direction ratios become direction cosines only after dividing by the magnitude.

Concept: Direction cosines of a vector
Typical timeTypical: 120 s
Your timeYours: —

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