Solveeit Logo
Login

Question

Mathematics Question on Vector Algebra

If ∣a⃗∣=15,∣b⃗∣=12|\vec{a} | = 15 , |\vec{b} | = 12 and ∣a⃗+b⃗∣=20|\vec{a} + \vec{b} | = 20 then ∣a⃗−b⃗∣=|\vec{a} - \vec{b} | =

A

338\sqrt{338}

B

338

C

769

D

769\sqrt{769}

Answer

338\sqrt{338}

Explanation

Solution

∣a⃗∣=15,∣b⃗∣=12,∣a⃗+b⃗∣=20\left|\vec{a}\right|=15, \left|\vec{b}\right|=12, \left|\vec{a} + \vec{b}\right| =20
∣a⃗+b⃗∣2=∣a⃗∣2+2a⃗.b⃗+∣b∣2\left|\vec{a} + \vec{b}\right|^{2} = \left|\vec{a}\right|^{2} +2 \vec{a}.\vec{b} +\left|b\right|^{2}
2a⃗.b⃗=(20)2−(15)2−(12)22 \vec{a} .\vec{b} =\left(20\right)^{2} -\left(15\right)^{2} -\left(12\right)^{2}
=400−225−144=31= 400 - 225 -144 = 31
Now, ∣a⃗−b⃗∣2=∣a⃗∣2−2a⃗.b⃗+∣b⃗∣2\left|\vec{a} - \vec{b}\right|^{2} =\left|\vec{a}\right|^{2} - 2\vec{a} . \vec{b} + \left|\vec{b}\right|^{2}
=(15)2−31+(12)2=225−31+144=338=\left(15\right)^{2} - 31 +\left(12\right)^{2} = 225 - 31 + 144 = 338
∴   ∣a⃗−b⃗∣338\therefore\:\:\: \left|\vec{a} - \vec{b}\right| \sqrt{338}