Question

. (a) Let a and b be two non-zero vectors with a common initial point 0 and whose directions are inclined at an angle O. Define the vector product of a and b. (b) The position vectors of the points A, B, C and D are 2i + j + 3k, 4i - j + 7k, -i - 2j - 2k and 2i - 5j + k respectively. Show that the lines AB and CD are skew. Find a direction

vector for their common perpendicular and determine the shortest distance between the two lines.

Answer 1

(a) Given that, a,b are two nonzero vectors wose directions are inclined at .

The vector product of a & b is denoted as    and it gives a vector quantity whose modulus value is and the direction is along the perpendicular to both vectors.

= where denoted as the unit vector perpendicular to both a & b.

(b) The position vectors of

A= 2i+j+3k, B = 4i-j+7k, C = -i-2j-2k , D = 2i-5j+k

then,

the equation of the line AB,

and the equation of line CD,

Any point of the line AB can be written as

Any point of the line CD can be written as

No such exists such that, =

Hence, AB, CD are skew.

Now, let the shortest distance line of AB and CD meets AB at E and CD at F.

Let, E=

and F=

Then, EF=

which is perpendicular to (2,-2,4) and (3,-3,3)

Then,

and

which gives

Then, the common perpendicular vector EF= (-3i-3j+0k)

and the shortest distance d= unit