If a⃗= (3,-1,-4), b⃗=(-2,4,-3) and c⃗=(-5,7,-1) findI) Unit vector along a⃗-2b⃗+c⃗ II) The direction cosines of the line represented by the vector a⃗-2b⃗+c⃗

If a⃗= (3,-1,-4), b⃗=(-2,4,-3) and c⃗=(-5,7,-1) find

I)
Unit vector along a⃗-2b⃗+c⃗

II)
The direction cosines of the line represented by the vector
a⃗-2b⃗+c⃗

1 Answers

Solution

Given vectors are

a⃗= (3,-1,-4), b⃗=(-2,4,-3) and c⃗=(-5,7,-1)

a⃗-2b⃗+c⃗ = (3,-1,-4) -2(-2,4,-3)+(-5,7,-1)

= (3,-1,-4) -(-4,8,-6)+(-5,7,-1)

={3-(-4)+(-5) , -1-8+7, -4-(-6)+(-1)}

= (2,-2, 1)

I) For unit vector

| a⃗-2b⃗+c⃗ |= =

II) For direction cosines of the line represented by a⃗-2b⃗+c⃗

Given, ABCD is a parallelogram

P is point of intersection of the diagonals

O is any point

Let OA,OB,OC,OD and OP be drawn.

Then we have,

Now,

( For a parallelogram P bisects the two diagonal.)

Given vectors are

a= 3i+j-k

b= xi-4j+4k

since they are collinear, the relation of their coordinates is given as:

Taking 1^st and 2^nd ratio we get,