Exercise 11.2

Show that the three lines with direction cosines 12 3 4 4 12 3 3 4 12, , ; , , ; , ,13 13 13 13 13 13 13 13 13 − − − are mutually perpendicular.

3 2 2

p / − − − = =

3 1 5

p − − − = = are at right angles.

7 5 1 x y z− + = =− and 1 2 3 x y z= = are perpendicular to each other.

Find the shortest distance between the lines ˆˆ ˆ( 2 )= + +/arrowrightnospr i j k + ˆˆ ˆ( )i j kλ − + and ˆ ˆˆ ˆ ˆ ˆ2 (2 2 )= − − + µ + +/arrowrightnospr i j k i j k

Find the shortest distance between the lines 1 1 1 7 6 1 x y z+ + += = − and 3 5 7 1 2 1 x y z− − − = =−

Find the shortest distance between the lines whose vector equations are ˆˆ ˆ( 2 3 )= + +/arrowrightnospr i j k + ˆˆ ˆ( 3 2 )i j kλ − + and ˆ ˆˆ ˆ ˆ ˆ4 5 6 (2 3 )= + + + µ + +/arrowrightnospr i j k i j k

Find the shortest distance between the lines whose vector equations are ˆˆ ˆ(1 ) ( 2) (3 2 )= − + − + −/arrowrightnospr t i t j t k and ˆˆ ˆ( 1) (2 1) (2 1)= + + − − +/arrowrightnospr s i s j s k

Show that the line through the points (1, - 1, 2), (3, 4, - 2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (- 1, - 2, 1), (1, 2, 5).

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector ˆˆ ˆ3 2 2i j k+ − .

Find the equation of the line in vector and in cartesian form that passes through the point with position vector ˆˆ2 4i j k− + and is in the direction ˆˆ ˆ 2i j k+ − .

Find the cartesian equation of the line which passes through the point (- 2, 4, - 5)

3 5 6 x y z+ − += = .

3 7 2 x y z− + −= = . Write its vector form.

Find the angle between the following pairs of lines: (i) ˆ ˆˆ ˆ ˆ ˆ2 5 (3 2 6 )= − + + λ + +/arrowrightnospr i j k i j k and ˆ ˆˆ ˆ ˆ7 6 ( 2 2 )= − + µ + +/arrowrightnospr i k i j k MATHEMATICS390 (ii) ˆ ˆˆ ˆ ˆ ˆ3 2 ( 2 )= + − + λ − −/arrowrightnospr i j k i j k and ˆ ˆˆ ˆ ˆ ˆ2 56 (3 5 4 )= − − + µ − −/arrowrightnospr i j k i j k

Find the angle between the following pair of lines: (i) 2 1 3 2 4 5 and2 5 3 1 8 4 x y z x y z− − + + − −= = = =− − (ii) 5 2 3and2 2 1 4 1 8 x y z x y z − − −= = = =