Question

The position vectors of points A, B, C and D are A =3ˆi+4ˆj+5kˆ r , B= 4ˆi+5ˆj+6kˆ , C = 7ˆi +9ˆj+3kˆ and D= 4ˆi+6ˆj , then displacement vectors AB and CD are

Answer 1

Answer:

displacement=r2-r1

AB=r2-r1 =4i+5j+6k-3i+4j+5k=1i+1j+1k=√3

CD=4i+6j-7i+9j+3k=-3i-2j+3k=√9+9+4=√22

hope so helpful

Answer 2

Displacement vector $\vec {AB}=\hat i+\hat j+\hat k$

Displacement vector $\vec {CD}=-3(\hat i+\hat j+\hat k)$

Explanation:

Given

Position vector of A

$\vec x_A=3\hat i+4\hat j+5\hat k$

Position vector of B

$\vec x_B=4\hat i+5\hat j+6\hat k$

Position vector of C

$\vec x_C=7\hat i+9\hat j+3\hat k$

Position vector of D

$\vec x_D=4\hat i+6\hat j$

Displacement vector $\vec {AB}$

$\vec{AB}=\vec x_B-\vec x_A$

$\implies \vec{AB}=(4\hat i+5\hat j+6\hat k)-(3\hat i+4\hat j+5\hat k)$

$\implies \vec{AB}=(4-3)\hat i+(5-4)\hat j+(6-5)\hat k$

$\implies \vec{AB}=\hat i+\hat j+\hat k$

Similarly,

Displacement vector $\vec {CD}$

$\vec{CD}=\vec x_D-\vec x_C$

$\implies \vec{CD}=(4\hat i+6\hat j)-(7\hat i+9\hat j+3\hat k)$

$\implies \vec{CD}=-3\hat i-3\hat j-3\hat k$

$\implies \vec{CD}=-3(\hat i+\hat j+\hat k)$

We can see that both vectors are anti-parallel because unit vectors along the direction of $\vec{AB}$ and $\vec {CD}$ are same while the direction is opposite due to negative sign.

Hope this answer is helpful.

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