The points A \left(\right. 4 , 5 , 10 \left.\right) , B \left(\right. 2 , 3 , 4 \left.\right) and C \left(\right. 1 , 2 , - 1 \left.\right) are three vertices of a parallelogram ABCD. Find the vector equations of the sides AB and BC and also find the coordinates of point D.

Engineering

Mathematics

Line of Greatest Slope

Question

The points

A (4, 5, 10), B (2, 3, 4)

and

C (1, 2, - 1)

are three vertices of a parallelogram ABCD. Find the vector equations of the sides AB and BC and also find the coordinates of point D.

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The points

A (4, 5, 10), B (2, 3, 4)

and

C (1, 2, - 1)

are three vertices of parallelogram ABCD.

Let coordinates of D be (x, y, z)
Direction vector along AB is

\vec{a} = (2 - 4) \hat{i} + (3 - 5) \hat{j} + (4 - 10) \hat{k} = - 2 \hat{i} - 2 \hat{j} - 6 \hat{k}

∴

Equation of line AB, is given by

\vec{b} = (4 \hat{i} + 5 \hat{j} + 10 \hat{k}) + λ (2 \hat{i} + 2 \hat{j} + 6 \hat{k})

Direction vector along BC is

\vec{c} = (1 - 2) \hat{i} + (2 - 3) \hat{j} + (- 1 - 4) \hat{k} = - \hat{i} - \hat{j} - 5 \hat{k}

∴

Equation of a line BC, is given by

\vec{d} = (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) + μ (\hat{i} + \hat{j} + 5 \hat{k})

Since ABCD is a parallelogram AC and BD bisect each other

\therefore \left,[ \dfrac{4 + 1}{2}, \dfrac{5 + 2}{2}, \dfrac{10 - 1}{2} \right ] =,\left [ \dfrac{2 + x}{2}, \dfrac{3 + y}{2}, \dfrac{4 + z}{2} \right ]

\Rightarrow 2 + x = 5, 3 + y = 7, 4 + z = 9

\Rightarrow x = 3, y = 4, z = 5

Coordinates of D are

(3, 4, 5) .

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