Question

Determine the value of M so that A vector = 2 i cap +m j cap + k cap amd B vector = 4 i cap -2j cap -2k cap are perpendicular​

Answer 1

Answer

Given -

$\mathrm{\vec{A} = 2\hat{\imath} + m\hat{\jmath} + \hat{k}}$

$\mathrm{\vec{B} = 4\hat{\imath} -2\hat{\jmath} -2 \hat{k}}$

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To find -

Value of m such that two vectors are perpendicular.

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Solution -

When the vectors are perpendicular the dot product of vectors is equal to 0.

Dot product for two vectors -

$\mathrm{\vec{A} • \vec{B} = (A_1 \hat{\imath} + A_2 \hat{\jmath} + A_3\hat{k} )(B_1\hat{\imath} + B_2\hat{\jmath} + B_3\hat{k})}$

$\boxed{\mathrm{\red{\vec{A} • \vec{B} = A_1B_1 + A_2B_2 + A_3B_3}}}$

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For given two vectors -

$\mathrm{\vec{A} = 2\hat{\imath} + m\hat{\jmath} +\hat{k}}$

$\mathrm{\vec{B} = 4\hat{\imath} -2\hat{\jmath} -2 \hat{k}}$

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$\mathrm\red{\vec{A} • \vec{B} = (2 \times 4) + (m \times (- 2)) + ( -2 ) = 0 }$

$\mathrm {8 - 2m - 2 = 0 }$

$\mathrm {6 - 2m = 0}$

$\mathrm {2m = 6}$

$\mathrm\pink{m = 3}$

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$\underline{\mathrm\red{Value\: of\: m\: for\: which </p><p>\:2 \:vectors \:are \:perpendicular\: is\: 3}}$