Let ABCD be the parallelogram whose sides ABand AD are represented by the vectors 2-4-5k- and -2-3k- respectively- Then- if a is a unit vector parallel to AC- then a is equal to -

Explanation for the correct option:Calculating the value of a:Let ABCD be the parallelogram.Given that,AB=2î+4ĵ 5k̂0ex0exAD=î+2ĵ+3k̂AC is the diagonal of a give ...

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Standard VIII

Mathematics

Construction of a Rhombus When One Side and One Angle Are Given.

Let ABCD be t...

Question

Let $A B C D$ be the parallelogram whose sides $\vec{A B}$ and $\vec{A D}$ are represented by the vectors $2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $\hat{i} + 2 \hat{j} + 3 \hat{k}$ respectively. Then, if $\vec{a}$ is a unit vector parallel to $\vec{A C}$ , then $\vec{a}$ is equal to ?

$\frac{3 \hat{i} - 6 \hat{j} - 2 \hat{k}}{3}$

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$\frac{3 \hat{i} + 6 \hat{j} + 2 \hat{k}}{3}$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

$\frac{3 \hat{i} - 6 \hat{j} - 2 \hat{k}}{7}$

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$\frac{3 \hat{i} + 6 \hat{j} - 2 \hat{k}}{7}$

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Solution

The correct option is D
$\frac{3 \hat{i} + 6 \hat{j} - 2 \hat{k}}{7}$

Explanation for the correct option:
Calculating the value of $\vec{a}$ :
Let $A B C D$ be the parallelogram.
Given that,
$\vec{A B} = 2 \hat{i} + 4 \hat{j} - 5 \hat{k} \vec{A D} = \hat{i} + 2 \hat{j} + 3 \hat{k}$
$\vec{A C}$ is the diagonal of a given parallelogram. Then by the parallelogram law, we get :
$\Rightarrow \vec{A C} = \vec{A B} + \vec{A D} = (2 \hat{i} + 4 \hat{j} - 5 \hat{k}) + (\hat{i} + 2 \hat{j} + 3 \hat{k}) = 3 \hat{i} + 6 \hat{j} - 2 \hat{k}$
The unit vector $\hat{a}$ parallel to $\vec{A C}$ is $\hat{a} = \frac{\vec{A C}}{|\vec{A C}|}$ .
Now,
$|\vec{A C}| = \sqrt{3^{2} + 6^{2} + (- 2^{2})} = \sqrt{49} = 7$
Therefore, $\vec{a} = \frac{3 \vec{i} + 6 \vec{j} - 2 \vec{k}}{7}$ .
Hence, option (D) is the correct answer.

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Let ABCD be the parallelogram whose sides AB→and AD→ are represented by the vectors 2i^+4j^-5k^ and i^+2j^+3k^ respectively. Then, if a→ is a unit vector parallel to AC→, then a→ is equal to ?

Let $A B C D$ be the parallelogram whose sides $\vec{A B}$ and $\vec{A D}$ are represented by the vectors $2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $\hat{i} + 2 \hat{j} + 3 \hat{k}$ respectively. Then, if $\vec{a}$ is a unit vector parallel to $\vec{A C}$ , then $\vec{a}$ is equal to ?