Question

If a =5i-j-3k and b=i+3j-5k, then show that the vectors a + b and a - b are perpendicular.

Answer 1

see below

Step-by-step explanation:

We need to show that the sum of the given two vectors and difference are perpendicular to each other. The given two vectors are,

a = 5i - j - 3k

b = i + 3j - 5k

Sum of these two vectors will be,

→ a + b = 5i + i - j + 3j - 3k - 5k

→ a + b = 6i + 2j - 8k

Difference of these two vectors will be,

→ a - b = 5i - i - j -3j -3k +5k

→ a - b = 4i - 4j + 2k

Now if two vectors are perpendicular then their dot product is zero.

So we can check if their dot product is zero or not as,

→ (a+b).(a-b) = ( 6i + 2j - 8k ) . ( 4i - 4j + 2k )

→ (a+b).(a-b) = 24 - 8 - 16

→ (a+b).(a-b) = 0

Hence here we can see that their dot product is zero . Hence we can say that the sum and difference of the given two vectors are perpendicular to each other.