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Applications of Dot Product

If a⃗,b⃗ an...

Question

If $\to a, \to b$ and $\to c$ be three vectors such that $| \to a | = 3, | \to b | = 4, | \to c | = 5$ and each one is perpendicular to the sum of the other two vectors, then find $| \to a + \to b + \to c |^{2}$ .

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Solution

Given:

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

Also,

Each one of them being perpendicular to the sum of the other two vectors

Therefore,

$\to a$ is perpendicular to $\to b + \to c$

$\Rightarrow \to a \cdot (\to b + \to c) = 0$

$\Rightarrow \to a \cdot \to b + \to a \cdot \to c = 0$ …… (1)

$\to b$ is perpendicular to $\to a + \to c$

$\Rightarrow \to b \cdot (\to a + \to c) = 0$

$\Rightarrow \to b \cdot \to a + \to b \cdot \to c = 0$ …… (2)

$\to c$ is perpendicular to $\to a + \to b$

$\Rightarrow \to c \cdot (\to a + \to b) = 0$

$\Rightarrow \to c \cdot \to a + \to c \cdot \to b = 0$ …… (3)

Now,

${∣ ∣ \to a + \to b + \to c ∣ ∣}^{2} = (\to a + \to b + \to c) \cdot (\to a + \to b + \to c)$

${∣ ∣ \to a + \to b + \to c ∣ ∣}^{2} = \to a \cdot \to a + \to a \cdot \to b + \to a \cdot \to c + \to b \cdot \to a + \to b \cdot \to b + \to b \cdot \to c + \to c \cdot \to a + \to c \cdot \to b + \to c \cdot \to c$

${∣ ∣ \to a + \to b + \to c ∣ ∣}^{2} = \to a \cdot \to a + (\to a \cdot \to b + \to a \cdot \to c) + \to b \cdot \to b + (\to b \cdot \to a + \to b \cdot \to c) + \to c \cdot \to c + (\to c \cdot \to a + \to c \cdot \to b)$

From equations (1), (2) and (3), we have

${∣ ∣ \to a + \to b + \to c ∣ ∣}^{2} = \to a \cdot \to a + (0) + \to b \cdot \to b + (0) + \to c \cdot \to c + (0)$

${∣ ∣ \to a + \to b + \to c ∣ ∣}^{2} = \to a \cdot \to a + \to b \cdot \to b + \to c \cdot \to c$

${∣ ∣ \to a + \to b + \to c ∣ ∣}^{2} = {| \to a |}^{2} + {∣ ∣ \to b ∣ ∣}^{2} + {| \to c |}^{2}$

${∣ ∣ \to a + \to b + \to c ∣ ∣}^{2} = 3^{2} + 4^{2} + 5^{2}$

${∣ ∣ \to a + \to b + \to c ∣ ∣}^{2} = 50$

So,
$∣ ∣ \to a + \to b + \to c ∣ ∣ = \sqrt{50}$

$∣ ∣ \to a + \to b + \to c ∣ ∣ = 5 \sqrt{2}$

Hence, this is the required result.

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