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Vector Algebra

Short Answer Type

151.

straight a with rightwards arrow on top space and space straight b with rightwards arrow on top

are two vectors such that

open vertical bar straight a with rightwards arrow on top close vertical bar space equals 2 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 3 space and space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 3 comma

find the angle between

straight a with rightwards arrow on top space and space straight b with rightwards arrow on top.

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152.
Find the angle between two vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ having the same length (magnitude) $square root of 2$ and their scalar products is -1.

Let $straight theta$ be angle between $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top.$
Now, $straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space minus 1$
$rightwards double arrow space space open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar space cos space straight theta space equals space minus 1 space space space space space space rightwards double arrow space space space left parenthesis square root of 2 right parenthesis thin space left parenthesis square root of 2 right parenthesis space cos space straight theta space equals space minus 1 space space space open square brackets because space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight b with rightwards arrow on top close vertical bar space equals square root of 2 close square brackets rightwards double arrow space space 2 space cos space straight theta space equals space minus 1 space space rightwards double arrow space space cos space straight theta space equals space minus 1 half space space rightwards double arrow space space space straight theta space equals space 120 degree space space space space space space space space open square brackets because space 0 space less or equal than space straight theta space less or equal than space 180 degree close square brackets$

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153.

Find the angle between two vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ such that $open vertical bar straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 3$ and $straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 1$ .

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154. Find the magnitude of two vectors

straight a with rightwards arrow on top space and space straight b with rightwards arrow on top

having the same magnitude and such that the angle between them is 30° and their scalar product is 3.

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155. Find the magnitude of two vectors

straight a with rightwards arrow on top space and space straight b with rightwards arrow on top

having the same magnitude and such that the angle between then is 60° and their scalar product is 1/2.

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156.

Find $open vertical bar straight a with rightwards arrow on top close vertical bar$ and $open vertical bar straight b with rightwards arrow on top close vertical bar comma$ if $open parentheses straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close parentheses. space left parenthesis straight a with rightwards arrow on top space minus straight b with rightwards arrow on top right parenthesis space equals space 12 space and space open vertical bar straight a with rightwards arrow on top close vertical bar space equals 2 space open vertical bar straight b with rightwards arrow on top close vertical bar.$

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157.

Find $open vertical bar straight a with rightwards arrow on top close vertical bar space and space space open vertical bar straight b with rightwards arrow on top close vertical bar space if space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space equals space 8 space space and space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 8 space open vertical bar straight b with rightwards arrow on top close vertical bar.$

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158.

Find $open vertical bar straight a with rightwards arrow on top close vertical bar space and space open vertical bar straight b with rightwards arrow on top close vertical bar space if space space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. thin space left parenthesis straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis space equals 3 space and space 2 space open vertical bar straight b with rightwards arrow on top close vertical bar space equals open vertical bar straight a with rightwards arrow on top close vertical bar.$

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159.

If $left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis space equals 0$ show that $open vertical bar straight a with rightwards arrow on top close vertical bar space equals open vertical bar straight b with rightwards arrow on top close vertical bar$

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160.

If $straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space straight a with rightwards arrow on top. space straight c with rightwards arrow on top comma space$ prove that either $straight a with rightwards arrow on top space equals space 0 with rightwards arrow on top$ or $straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top$ or $straight a with rightwards arrow on top$ is perpendicular to $straight b with rightwards arrow on top minus straight c with rightwards arrow on top$ .