Question

L defined by the following Question 9 What is the relationship between the vector (1,2,-2) and the line? parametric equations 2= 3+t y = 1 + 2t te R. 2= 1-2t Verify that 2-3-4-1-+1 is an equation of the line.

Answer 1

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We find that the given equation is in parametric form:

$x=3+t\implies \frac{x-3}{1}=t \\y=1+2t \implies \frac{y-1}{2}=t\\z=1-2t\implies \frac{z-1}{-2}=t\\$

Thus we have

$\frac{x-3}{1}= \frac{y-1}{2}= \frac{z-1}{-2}=t\\ \frac{x-3}{1}= \frac{y-1}{2}= \frac{z-1}{-2}$

which is the required equation of the line of the line $l$.

Relation between vector and Line :

The vector $(1,2,-2)$ are the direction ratios of the line $l$ .

The direction ratios are proportional to the direction cosines of the line $l$ .

By direction cosines we mean the cosine of the angle the  line $l$ makes with the co-ordinate axes $x,y$ and $z$.

Also note that from the given equation we find that $(1,2,-2)$ are direction ratios of the given line