(B) The direction cosines of a vector $\vec{a} = x \hat{i} + y \hat{j} + z \hat{k}$ are given by $\frac{x}{|\vec{a}|}, \frac{y}{|\vec{a}|}, \frac{z}{|

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$\frac{-2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{-5}{\sqrt{30}}$

(B) The direction cosines of a vector $\vec{a} = x \hat{i} + y \hat{j} + z \hat{k}$ are given by $\frac{x}{|\vec{a}|}, \frac{y}{|\vec{a}|}, \frac{z}{|\vec{a}|}$.Given $\vec{a} = -2 \hat{i} + \hat{j} - 5 \hat{k}$.First,calculate the magnitude of the vector: $|\vec{a}| = \sqrt{(-2)^2 + (1)^2 + (-5)^2} = \sqrt{4 + 1 + 25} = \sqrt{30}$.The direction cosines are $\frac{-2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{-5}{\sqrt{30}}$.Thus,option $B$ is correct.

Class 12 Mathematics Vector Algebra Basic , Modulus and Algebra of vectors

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The direction cosines of the vector $\vec{a} = -2 \hat{i} + \hat{j} - 5 \hat{k}$ are

AP EAMCET 2020 All AP EAMCET

A
$\frac{-2}{\sqrt{8}}, \frac{1}{\sqrt{8}}, \frac{-5}{\sqrt{8}}$
B
$\frac{-2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{-5}{\sqrt{30}}$
C
$\frac{2}{\sqrt{8}}, \frac{-1}{\sqrt{8}}, \frac{5}{\sqrt{8}}$
D
$\frac{-2}{\sqrt{30}}, \frac{-1}{\sqrt{30}}, \frac{-5}{\sqrt{30}}$

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Let $\vec{a}, \vec{b},$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear. If the vector $\vec{a} + 2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b} + 3\vec{c}$ is collinear with $\vec{a}$,then $\vec{a} + 2\vec{b} + 6\vec{c} = \dots$

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Let $A$ and $B$ be points with position vectors $a$ and $b$ with respect to the origin $O$. If the point $C$ on $OA$ is such that $2AC = CO$,$CD$ is parallel to $OB$ and $|\overrightarrow{CD}| = 3|\overrightarrow{OB}|$,then $\overrightarrow{AD}$ is equal to

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If a parallelogram is constructed on the vectors $\vec{a} = 3\vec{p} - \vec{q}$ and $\vec{b} = \vec{p} + 3\vec{q}$, where $|\vec{p}| = 3$, $|\vec{q}| = 2$ and the angle between $\vec{p}$ and $\vec{q}$ is $\pi/3$, then the ratio of the lengths of adjacent sides $|\vec{a}|$ and $|\vec{b}|$ of the parallelogram is:

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If $a$ is collinear with $b = 3 \hat{i} + 6 \hat{j} + 6 \hat{k}$ and $a \cdot b = 27$,then $|a| =$

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$I$. Two non-zero, non-collinear vectors are linearly independent.
$II$. Any three coplanar vectors are linearly dependent.
Which of the above statements is/are true?

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The direction cosines of the vector $\vec{a} = -2 \hat{i} + \hat{j} - 5 \hat{k}$ are

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Let $\vec{a}, \vec{b},$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear. If the vector $\vec{a} + 2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b} + 3\vec{c}$ is collinear with $\vec{a}$,then $\vec{a} + 2\vec{b} + 6\vec{c} = \dots$

Let $A$ and $B$ be points with position vectors $a$ and $b$ with respect to the origin $O$. If the point $C$ on $OA$ is such that $2AC = CO$,$CD$ is parallel to $OB$ and $|\overrightarrow{CD}| = 3|\overrightarrow{OB}|$,then $\overrightarrow{AD}$ is equal to

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$I$. Two non-zero, non-collinear vectors are linearly independent.
$II$. Any three coplanar vectors are linearly dependent.
Which of the above statements is/are true?