1.

The lines having directional derivatives as a1a2+b1b2+c1c2=0​

Answer»

Answer:

1. SHOW that the three lines with direction cosines

Are mutually perpendicular.

Solution:

Let us consider the direction cosines of L1, L2 and L3 be l1, m1, N1; l2, m2, n2 and l3, M3, n3.

We know that

If l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines;

And θ is the acute angle between the two lines;

Then COS θ = |l1l2 + m1m2 + n1n2|

If two lines are perpendicular, then the angle between the two is θ = 90°

For perpendicular lines, | l1l2 + m1m2 + n1n2 | = cos 90° = 0, i.e. | l1l2 + m1m2 + n1n2 | = 0

So, in order to check if the three lines are mutually perpendicular, we compute | l1l2 +

m1m2 + n1n2 | for all the pairs of the three lines.

Firstly let us compute, | l1l2 + m1m2 + n1n2 |

So, L1⊥ L2 …… (1)

Similarly,

Let us compute, | l2l3 + m2


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