Question

Abar ×b bar =c bar×d bar and abar×cbar =bbar ×d bar mode of abar not equal to mode of dbar and mode of b bar not equal to mode of c bar then show that (a bar -b bar ) and (bbar -cbar) are parallel

Answer 1

Condition of parallelism

If the two vectors $\mathrm{\vec{a}}$ and $\mathrm{\vec{b}}$ parallel to each other, then

• $\mathrm{\vec{a}\times\vec{b}=\vec{0}}$

Given:

• $\mathrm{\vec{a}\times\vec{b}=\vec{c}\times\vec{d}}$

• $\mathrm{\vec{a}\times\vec{c}=\vec{b}\times\vec{d}}$

• $\mathrm{|\vec{a}|\neq |\vec{d}|,\:|\vec{b}|\neq |\vec{c}|}$

To prove:

• $\mathrm{(\vec{a}-\vec{d})}$ and $\mathrm{(\vec{b}-\vec{c})}$ are parallel

Proof:

Now $\mathrm{(\vec{a}-\vec{d})\times (\vec{b}-\vec{c})}$

$\mathrm{=\vec{a}\times\vec{b}-\vec{a}\times\vec{c}-\vec{d}\times\vec{b}+\vec{d}\times\vec{c}}$

$\mathrm{=\vec{a}\times\vec{b}-\vec{a}\times\vec{c}+\vec{b}\times\vec{d}-\vec{c}\times\vec{d}}$

$\mathrm{=\vec{a}\times\vec{b}-\vec{a}\times\vec{c}+\vec{a}\times\vec{c}-\vec{a}\times\vec{b}}$

$\mathrm{=\vec{0}}$

This satisfy the condition of parallelism.

Thus $\mathrm{(\vec{a}-\vec{d})}$ and $\mathrm{(\vec{b}-\vec{c})}$ are parallel.

This completes the proof.

Answer 2

Answer:

abcd

Step-by-step explanation: