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Test for Collinearity of Vectors

Let a⃗, b⃗, c...

Question

Let $\to a, \to b, \to c$ are three unit vectors such that no two of the vectors are collinear. If the vector $\to a + \to b$ is collinear with $\to c$ and the vector $\to b + \to c$ is collinear with $\to a,$ then the value of $| \to a + \to b + \to c |$ is

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Solution

$\to a + \to b = x \to c$
$\to b + \to c = y \to a$
Now, $\to a + \to b + \to c = (1 + x) \to c \dots (1)$
and $\to a + \to b + \to c = (1 + y) \to a \dots (2)$
From $(1) - (2)$
$(1 + x) \to c - (1 + y) \to a = 0$
$\Rightarrow m \to c - n \to a = 0$
From here $\to c, \to a$ are collinear but as per the question $\to c, \to a$ are non-collinear.
$∴ x = - 1; y = - 1$
$∴ | \to a + \to b + \to c | = 0$

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Test for Collinearity of Vectors

Standard XII Mathematics

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