Question

# Suppose the direction cosines of two lines are given by al+bm+cn=0 and fmn+gln+hlm=0 where f, g, h, a, b, c are arbitrary constants and l, m, n are direction cosines of the lines. On the basis of the above information answer the following The given lines will be perpendicular if

Hint:

### Direction ratios of the line are quantities that are proportional to the direction cosines of the line. We have taken the directional cosines of the lines to be l, m, and n. Here we have given the direction cosines of two lines are given by al+bm+cn=0 and fmn+gln+hlm=0 where f, g, h, a, b, c are arbitrary constants and l, m, n are direction cosines of the lines. We have to find when will be the given lines be perpendicular.

## The correct answer is:

### We are aware that the direction cosine is equal to the cosine of the angle formed by the line intersecting each of the three coordinate axes, namely the x, y, and z axes. If the angles subtended by these three axes are α, β, and γ, then the direction cosines are cos α, cos β, cos γ respectively.

We know that distance of (x, y,z) from origin is r: x^{2} + y^{2} + z^{2} = r^{2}So: l^{2} + m^{2} + n^{2} = 1

Now that we have given: al+bm+cn=0 and fmn+gnl+hlm=0. Canceling n, we get:

We define lines using cosine ratios of the line. While working with three-dimensional geometry (used in so many applications such as game designing), it is needed to express the importance of the line present in 3-D space. Here we were asked to find the condition for perpendicular line, so it is .

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### Suppose the direction cosines of two lines are given by al+bm+cn=0 and fmn+gln+hlm=0 where f, g, h, a, b, c are arbitrary constants and l, m, n are direction cosines of the lines. On the basis of the above information answer the following For f=g=h=1 both lines satisfy the relation

All the above options are correct.

### Suppose the direction cosines of two lines are given by al+bm+cn=0 and fmn+gln+hlm=0 where f, g, h, a, b, c are arbitrary constants and l, m, n are direction cosines of the lines. On the basis of the above information answer the following For f=g=h=1 both lines satisfy the relation

All the above options are correct.

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