Question

Let u, v and w be vectors such that u + v + w = 0. If |u| = 3 |v| = 4 and |w| = 5, then u.V + v.W + w.U is equal to

Answer 1

Let u, v and w be vectors such that u + v + w = 0. If |u| = 3 |v| = 4 and |w| = 5, then u.v + v.w + w.u is equal to

it is given that, u + v + w = 0

⇒(u + v + w)² = 0

we know, (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

⇒u² + v² + w² + 2(u.v + v.w + w.u) = 0

⇒2(u.v + v.w + w.u) = -(|u|² + |v|² + |w|²)

putting the values of |u|, |v| and |w|,

⇒2(u.v + v.w + w.u) = -(3² + 4² + 5²)

⇒2(u.v + v.w + w.u) = -(9 + 16 + 25) = -50

⇒u.v + v.w + w.u = -50/2 = -25

hence, u.v + v.w + w.u = -25

Answer 2

Answer: -25

Step-by-step explanation:

Given,

∣u∣=3,∣v∣=4 and ∣w∣=5

Also, u+v+w=0

On squaring both sides, we get

∣u∣ 2+∣v∣ 2+∣w∣ 2+2(u.v+v.w+w.u)=0

⇒3 ^2 +4 ^2+5 ^2 +2(u.v+v.w+w.u)=0

⇒9+16+25+2(u.v+v.w+w.u)=0

⇒u.v+v.w+w.u=-25

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