Question

Express the vector vector a=5i-2j+5k as the sum of two vectors such that one is parallel to the vector b=3i+k and other is perpendicular to vector b

Answer 1

Answer: : a⇀ =5 i^ + 2j^ + 5 k^ , b⇀ = 3i^+ k^

Let Vector a⇀ = a1⇀ + a2⇀, Where a1⇀is parallel to b⇀ and a2⇀ is perpendicular to b⇀.

as a1⇀ is parallel to b⇀a1⇀ = λ(3i⇀+k⇀)=> 3λi⇀+ λk⇀ ................... (1)

Also

a⇀ = a1⇀ + a2⇀,

=> a2⇀ = a⇀ - a1⇀,

=> a2 ⇀= 5i⇀+2j⇀+5k⇀ - 3λi⇀-λk⇀

=> a2 ⇀ = (5-3λ)i⇀ + 2j⇀+ (5-λ)k⇀ .................(2)

As a2⇀ is perpendicular to b⇀

So, a2⇀·b⇀ = 0

((5-3λ)i⇀ + 2j⇀+ (5-λ)k⇀) . (3i⇀+k⇀) =0=> 15 - 9λ + 5 -λ = 0=> 10λ = 20 =>λ = 2

Put value of λ in both equations.

a1⇀ = 6i⇀+ 2k⇀

a2⇀ = -i⇀-2j⇀+3k

Step-by-step explanation:

Answer 2

Answer:

Step-by-step explanation:

Hole it helps u !!!!!!