If lines (x-1)/(-3)=(y-2)/(2 k)=(z-3)/(2) and (x-1)/(3 k)=(y-5)/(1)=(z-6)/(-5) are mutually perp

step 1 Hello. So today our question is that if the lines x plus 1 upon k plus 2 equals to y minus 1 upo

If lines (x-1)/(-3)=(y-2)/(2 k)=(z-3)/(2) and (x-1)/(3...

If lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\frac{x-1}{3 k}=\frac{y-5}{1}=\frac{z-6}{-5}$ are mutually perpendicular, then $\mathrm{k}$ is equal to-
(a) $\frac{-10}{7}$
(b) $-\frac{7}{10}$
(c) -10
(d) -7

Key Concepts

Condition for Orthogonality

Two vectors are said to be orthogonal or perpendicular when the dot product between them is zero. This condition is essential in many areas of mathematics and physics, particularly when determining perpendicular relationships between lines, planes, or higher-dimensional elements.

Dot Product of Vectors

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. It is fundamental in vector analysis because it provides a way to compute the angle between two vectors. When the dot product is zero, the vectors are orthogonal, meaning they meet at a right angle.

Direction Vector

A direction vector defines the orientation of a line in space. It is extracted directly from the denominators in the symmetric equation for a line, and is crucial for understanding how lines are positioned relative to one another. The direction vector plays a key role in determining relationships such as parallelism and perpendicularity.

Symmetric Form of a Line

The symmetric form is a way to represent a straight line in space using ratios that relate the differences in coordinates to the direction components. It encapsulates both a point on the line and a direction vector, providing a compact representation that is frequently used in analytic geometry.

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