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Applications of Dot Product

If a=3,b=4,...

Question

If $a = 3, b = 4, c = 5$ each one of $\to a, \to b$ and $\to c$ is perpendicular to the sum of the remaining then $∣ ∣ \to a + \to b + \to c ∣ ∣$

A

\frac{5}{\sqrt{2}}

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B

\frac{2}{\sqrt{5}}

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C

5 \sqrt{2}

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D

\sqrt{5}

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Solution

The correct option is C $5 \sqrt{2}$
It is given that,
$\to a \cdot (\to b + \to c) = 0 \Rightarrow \to a \cdot \to b + \to a \cdot \to c = 0$ .........(1)
$\to b \cdot (\to c + \to a) = 0 \Rightarrow \to b \cdot \to c + \to a \cdot \to b = 0$ .......(2)
and $\to c \cdot (\to a + \to b) = 0 \Rightarrow \to c \cdot \to a + \to c \cdot \to b = 0$ .......(3)
Add above three equations $\Rightarrow \to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a = 0$
Therefore
$| \to a + \to b + \to c | = \sqrt{| \to a |^{2} + | \to b |^{2} + | \to c |^{2} + 2 (\to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a)}$
$= \sqrt{3^{2} + 4^{2} + 5^{2} + 0} = \sqrt{50} = 5 \sqrt{2}$

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Similar questions

| \to a | = 3, ∣ ∣ \to b ∣ ∣ = 4, | \to c | = 5, \to a ⊥ (\to b + \to c), \to b ⊥ (\to c + \to a)

\to c ⊥ (\to a + \to b)

\sqrt{2} ∣ ∣ \to a + \to b + \to c ∣ ∣

\to a, \to b, \to c

be three vectors such that

| \to a | = 3, | \to b | = 4

| \to c | = 5

and each one of the perpendicular to the sum of the other two, find

| \to a + \to b + \to c |

\to a

\to b

\to c

be three vectors such that

| \to a | = 3 ∣ ∣ \to b ∣ ∣ = 4 | \to c | = 5

and each one of them being perpendicular to the sum of the other two find

∣ ∣ \to a + \to b + \to c ∣ ∣

Q. Magnitudes of vectors

\to a, \to b, \to c

3, 4, 5

respectively. If

\to a

\to b + \to c

\to b

\to c + \to a

\to c

\to a + \to b

are mutually perpendicular, then the magnitude of

\to a + \to b + \to c

$if | \to a | = 3, | \to b | = 4, | \to c | = 5, \to a ⊥ t o (\to b + \to c), \to b ⊥ t o (\to c + \to a) and \to c ⊥ t o (\to a + \to b), then | \to a + \to b + \to c | =$

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