Question

Let a = 2i + 4j -5k, b = i + j + k and c = j +2k. Find the unit vector in the opposite direction of a + b + c.

Answer 1

Given that

a = 2i + 4j -5k, b = i + j + k and c = j +2k

Then

a + b + c = 2i + 4j -5k + i + j + k + j +2k

a + b + c = 3 i + 6 j - 2 k

|a + b + c| = $\sqrt{3^{2}+6^{2}+(-2)^{2}}=\sqrt{49}=7$

unit vector in direction of a+b+c =  (3 i + 6 j - 2 k)/7

unit vector in direction of a+b+c =  3/7 i + 6/7 j - 2/7 k

AND

unit vector in opposite direction of a+b+c = - ( 3/7 i + 6/7 j - 2/7 k )

⇒

unit vector in opposite direction of a+b+c = - 3/7 i - 6/7 j + 2/7 k )

Answer 2

Answer:

unit vector in the opposite direction

$\frac{1}{7}(-3\hat i-6\hat j+2\hat k)$

Step-by-step explanation:

To find the unit vector in the direction of $\vec a+\vec b+\vec c$

if

$\vec a=2\hat i+4\hat j-5\hat k\\\\\vec b=\hat i+\hat j+\hat k\\\\\\ \vec c= \hat j+2\hat k$

$\vec a+\vec b+\vec c=3\hat i+6\hat j-2\hat k$

$|\vec a+\vec b+\vec c|=\sqrt{9+36+4}\\\\\\=7$

unit vector in direction of $\vec a+\vec b+\vec c$ is

=$\frac{1}{7}(3\hat i+6\hat j-2\hat k)$

unit vector inthe opposite direction of $\vec a+\vec b+\vec c$ is

=$\frac{1}{7}(-3\hat i-6\hat j+2\hat k)$