Question

1. Find a vector equation and parametric equations for a line passing through (-1,2,3) in the direction of Ŭ = i + 21 – R.

Answer 1

( and parametric equation for (-1,2,3) in the direction of Find en rector equation a line passing through = [ +29+ (-Å). Sol":- Vector equation in The position vector of the point (-1,2,3) is given by - î +2 +3ů. kit nity) tz û be the position vector of any point on the required line.. Then (hityſt zů) - (-êt2} +3%) is the vector along the required line. since the given line is in the direction of 3= itzg-ks so (alty] + z ů) - (-i+2j +3h) is parallel to the vector 3 = îtzŷ ê. Therefore their cross - product will be zerovector. That is, ((altyjt zh) - (-itz + 3k) x ([ + 25 - 2) = 0 suppose, r = nity3 + zů a = – l +2/+34 8 = C +2j- î (given) Thus the vector equation can be written an per a x 2 = 0, where sia and to as above.

Parametric Equation: det r be any point on the required wine. As the above reason, "roo(-1,2,3) "is parallel to the vector (1,2-D. [:D = î+zj - î as point i = (1, 2, -1)] Therefore one can be written as scalar multiple of other, ie, - (-1,2,3) = x (1,2,-1) XER lor, r = (1,2,3) + ♡ (112,-1) or, r = |x-1,2x+2, 3-d). Here ir gives the position of any arbitrang pon on the line if a varies on real numbers. Thus the parametrio equation with parameter & ER of the line is given by .. 1R 3] (d) = (2+, 24+2, 3-d). [VIR LER.