Question

If $2i+4j-5k$ and $i+2j+3k$ are adjacent sides of a parallelogram, then the lengths of its diagonals are:

• A. $7,\sqrt{69}$
• B. $6,\sqrt{59}$
• C. $5,\sqrt{65}$
• D. $5,\sqrt{55}$

Answer 1

The correct option is A

$7,\sqrt{69}$

Explanation for the correct answer:

Step 1: Gathering information about the parallelogram given

Let $ABCD$ be the parallelogram with sides $AD$ and $BC$ as $i+2j+3k$ and sides $DC$ and $AB$ as $2i+4j-5k$.

Also, let $2i+4j-5k=a$ and

$i+2j+3k=b$

Step 2: Applying the triangle law of addition

According to the triangle law of vector addition, the third vector taken in a different order is equal to the sum of the other two vectors taken in the same order.

From the diagram, $AC$ and $BD$ act as the diagonals of the parallelogram.

Now,

$AC=a+b$

$\Rightarrow$$AC=2i+4j-5k+i+2j+3k$

$\Rightarrow$$AC=3i+6j-2k$

Therefore, the length of $AC$ is given as

$\left|AC\right|=\left|3i+6j-2k\right|$

$\Rightarrow$$\left|AC\right|=\sqrt[]{9+36+4}$

$\Rightarrow$$\left|AC\right|=7$

Similarly,

$BD=b-a$

$\Rightarrow$$BD=i+2j+3k-(2i+4j-5k)$

$\Rightarrow$$BD=-i-2j+8k$

Therefore, the length of $AC$ is given as

$\left|BD\right|=\left|-i-2j+8k\right|$

$\Rightarrow$$\left|BD\right|=\sqrt[]{1+4+64}$

$\Rightarrow$$\left|BD\right|=\sqrt[]{69}$

Therefore, the length of diagonals $AC$ and $BD$ is $7,\sqrt{69}$ respectively.

Hence, Option (A) is the correct option.