If a- and b- are two non-zero and non-collinear vectors- then a- - b- and a- - b- are

The correct option is C linearly independent vectorsSince, a⃗ and b⃗ are two non zero non collinear nbsp;vectors nbsp;i.e. xa⃗+yb⃗=0 ⇒ x=0 ...

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Standard XII

Mathematics

Condition for Coplanarity of Four Points

If a⃗ and ...

Question

If $\to a$ and $\to b$ are two non-zero and non-collinear vectors, then $\to a + \to b$ and $\to a - \to b$ are

linearly dependent vectors

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linearly independent vectors

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linearly dependent and independent vectors

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none of the above

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Solution

The correct option is C linearly independent vectors
Since, $\to a$ and $\to b$ are two non-zero non-collinear vectors
i.e. $x \to a + y \to b = 0 \Rightarrow x = 0, y = 0$
Hence, $\to a$ and $\to b$ are linearly independent.
Now consider, $x_{1} (\to a + \to b) + x_{2} (\to a - \to b) = 0$
$(x_{1} + x_{2}) \to a + (x_{1} - x_{2}) \to b = 0$
$\Rightarrow x_{1} + x_{2} = 0$ ; $x_{1} - x_{2} = 0$
$\Rightarrow x_{1} = x_{2} = 0$
Hence, $\to a + \to b$ and $\to a - \to b$ are linearly independent.

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If →a and →b are two non-zero and non-collinear vectors, then →a+→b and →a−→b are

If $\to a$ and $\to b$ are two non-zero and non-collinear vectors, then $\to a + \to b$ and $\to a - \to b$ are