Three Dimensional Geometry — Practice

Question 1

If a line makes angles 90, 135 and 45 degrees with the x, y and z-axes, find its direction cosines.

Accepted Answer

Direction cosines are the cosines of the direction angles, so evaluate cos(90), cos(135), cos(45). Answer: (0, -1/sqrt2, 1/sqrt2).

Question 2

If a line has direction ratios -18, 12, -4, what are its direction cosines?

Accepted Answer

Since sqrt(324+144+16)=22, divide each direction ratio by 22 to normalise. Answer: (-9/11, 6/11, -2/11).

Question 3

Find the vector and Cartesian equations of the line through the point (2,-1,4) in the direction i+2j-k.

Accepted Answer

Substituting a=(2,-1,4) and b=(1,2,-1) into r=a+lambda.b gives the vector form, and the Cartesian form follows from the point and direction ratios. Answer: r=(2i-j+4k)+lambda(i+2j-k); (x-2)/1=(y+1)/2=(z-4)/(-1).

Question 4

Find the angle between the lines with direction ratios (2,2,1) and (4,1,8).

Accepted Answer

Since |8+2+8|=18 and the magnitudes are 3 and 9, cos(theta)=18/27. Answer: cos^-1(2/3).

Question 5

Show that the points (2,3,4), (-1,-2,1) and (5,8,7) are collinear.

Accepted Answer

The direction ratios of AB are (-3,-5,-3) and of BC are (6,10,6), and since BC=-2.AB with B common to both, all three points lie on one line. Answer: Collinear.

Question 6

Find the shortest distance between r=(i+2j+k)+lambda(i-j+k) and r=(2i-j-k)+mu(2i+j+2k).

Accepted Answer

Since b1 x b2=(-3,0,3) with magnitude 3sqrt2, and a2-a1=(1,-3,-2), the numerator is |−3+0−6|=9. Answer: 9/(3sqrt2) = 3/sqrt2 units.

Question 7

Find p so that (1-x)/3=(7y-14)/(2p)=(z-3)/2 and (7-7x)/(3p)=(y-5)/1=(6-z)/5 are perpendicular.

Accepted Answer

Rewriting into standard form gives direction ratios (-3, 2p/7, 2) and (-3p/7, 1, -5); setting the dot product to zero gives 9p/7+2p/7-10=0. Answer: p=70/11.

Question 8

Find the shortest distance between (x+1)/7=(y+1)/(-6)=(z+1)/1 and (x-3)/1=(y-5)/(-2)=(z-7)/1.

Accepted Answer

With a1=(-1,-1,-1), b1=(7,-6,1), a2=(3,5,7), b2=(1,-2,1), applying the skew formula |(b1 x b2).(a2-a1)|/|b1 x b2| gives the distance. Answer: 2.sqrt(29) units.

Practice — Three Dimensional Geometry

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