If A(2,1, 3), B(2, -3, 4) , C(-1, 2 , 7) are the three points ; find a point D such that |\overrightarrow { AB } |=\frac { 1 }{ 2 } |\overrightarrow { CD } | \, and \, AB||CD.

Engineering

Mathematics

Line of Greatest Slope

Question

If A(2,1, 3), B(2, -3, 4) , C(-1, 2 , 7) are the three points ; find a point D such that

|\overrightarrow { AB } |=\frac { 1 }{ 2 } |\overrightarrow { CD } |&nbsp;\, and \, AB||CD.

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\begin{array}{l} \vec{A B} = \frac{1}{2} \vec{C D} o r \vec{A B} = - \frac{1}{2} \vec{C D}, \\ \vec{A B} = - 4 \hat{j} + \hat{k}, \\ 2 (- 4 \hat{j} + \hat{k},) = \vec{D} - \vec{C} 2 (- 4 \hat{j} + \hat{k},) = \vec{C} - \vec{D}, \\ - 8 \hat{j} + 2 \hat{k} - \hat{i} + 2 \hat{j} + 7 \hat{k} = \hat{0} - 8 \hat{j} + 2 \hat{k} + \hat{i} - 2 \hat{j} - 7 \hat{k} = \hat{0}, \\ \vec{D} = - \hat{i} - 6 \hat{j} + 9 \hat{k} - \hat{i} - 10 \hat{j} - 5 \hat{k} = - \hat{0}, \\ o r \vec{D} = - \hat{i} + 10 \hat{j} + 5 \hat{k}, \end{array}

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