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Property 4

Let u, → v an...

Question

Let $\vec{u,} \vec{v} and \vec{w}$ be vectors such $\vec{u} + \vec{v} + \vec{w} = \vec{0} .$ If $|\vec{u}| = 3, |\vec{v}| = 4 and |\vec{w}| = 5,$ then find $\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} .$

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Solution

$Given that \vec{u} + \vec{v} + \vec{w} = 0 \Rightarrow |\vec{u} + \vec{v} + \vec{w}| = 0 \Rightarrow {|\vec{u} + \vec{v} + \vec{w}|}^{2} = 0 \Rightarrow {|\vec{u}|}^{2} + {|\vec{v}|}^{2} + {|\vec{w}|}^{2} + 2 (\vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u}) = 0 \Rightarrow 3^{2} + 4^{2} + 5^{2} + 2 (\vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u}) = 0 (Given : |\vec{u}| = 3, |\vec{v}| = 4 and |\vec{w}| = 5) \Rightarrow 9 + 16 + 25 + 2 (\vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u}) = 0 \Rightarrow 50 + 2 (\vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u}) = 0 \Rightarrow 2 (\vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u}) = - 50 ∴ \vec{u} . \vec{v} + \vec{v} . \vec{w} + \vec{w} . \vec{u} = \frac{- 50}{2} = - 25$

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u, v

w

be vectors such that

u + v + w = 0

| u | = 3, | v | = 4

| w | = 5

u . v + v . w + w . u

\to u, \to v

\to w

be vectors such that

\to u + \to v + \to w = \to 0 .

| \to u | = 3, | \to v | = 4

| \to w | = 5,

\to u . \to v + \to v . \to w + \to w . \to u

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be vectors such that

\to u + \to v + \to w = \to 0 .

| \to u | = 3, | \to v | = 4

| \to w | = 5,

\to u . \to v + \to v . \to w + \to w . \to u

Let u,v and w be vectors such that u+v+w. If |u|=3,|v|=4 and ||w|=5. Then the value of u.v+v.w+w.u is

\to u, \to v

\to w

be vectors such that

\to u + \to v + \to w = \to 0

| \to u | = 3, | \to v | = 4

| \to w | = 5

, then the value of

\to u . \to v + \to v . \to w + \to w . \to u

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