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Vector Algebra

Short Answer Type

141.

If $straight a with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 2 straight j with hat on top space plus space 9 straight k with hat on top space and space straight b with rightwards arrow on top space equals space straight i with hat on top space plus space straight lambda straight j with hat on top space plus space 3 straight k with hat on top comma$ find the value of $straight lambda$ so that $straight a with rightwards arrow on top plus straight b with rightwards arrow on top$ is perpendicular to $straight a with rightwards arrow on top minus straight b with rightwards arrow on top$ .

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142.

Let $straight a with rightwards arrow on top space equals space 5 straight i with hat on top space minus space straight j with hat on top space plus space 7 straight k with hat on top$ and $straight b with rightwards arrow on top space equals straight i with hat on top minus straight j with hat on top plus straight lambda straight k with hat on top.$ Find $straight lambda$ such that $straight a with rightwards arrow on top plus straight b with rightwards arrow on top$ and $straight a with rightwards arrow on top minus straight b with rightwards arrow on top$ are orthogonal .

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143.

Find $straight lambda$ if the vectors $5 straight i with hat on top plus 2 straight j with hat on top minus straight k with hat on top$ and $straight lambda straight i with hat on top space minus space straight j with hat on top space plus space 5 space straight k with hat on top$ are orthogonal.

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144.

If $straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals negative straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top space space and space straight c with rightwards arrow on top space equals 3 straight i with hat on top space plus space straight j with hat on top$ such that $straight a with rightwards arrow on top space plus space straight lambda straight b with rightwards arrow on top$ is perpendicular to $straight c with rightwards arrow on top$ , then find the value of $straight lambda.$

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Long Answer Type

145. Find the value of A such that the vectors

3 straight a with rightwards arrow on top plus space 4 straight b with rightwards arrow on top

and

2 straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top

are perpendicular to each other when

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Short Answer Type

146.

If $open vertical bar straight a with rightwards arrow on top close vertical bar space equals space square root of 3 comma space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 2 space and space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 3 comma$ 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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147. Find the angle between two vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ with magnitude 1 and 2 respectively and such that $straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 1$

Here,
a = 1, b = 2, $straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 1$
Let $straight theta$ be the angle between two vectors.
Now, $straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 1 space space space space space space rightwards double arrow space space ab space cos space straight theta space equals space 1 space space space space rightwards double arrow space space space left parenthesis 1 right parenthesis thin space left parenthesis 2 right parenthesis space cos space straight theta space equals space 1$
$therefore space space cos space straight theta space equals space 1 half space space space space space space rightwards double arrow space space space straight theta space equals space 60 degree$

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148.

If $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ are two vectors such that $open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 4 comma space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 3 space space and space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space equals space 6$ 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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149.

Find the angle between two vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ with magnitudes 2 and 1 respectively, and such that $straight a with rightwards arrow on top space. space straight b with rightwards arrow on top space equals space square root of 3.$

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150.

Find the angle between two vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ with magnitudes $square root of 3 space and space 2 comma space respectively comma space and space that space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space square root of 6.$