Let a=(1,2,3,4), b=(4,3,2,1) and c=(1,1,1,1) be vectors in R4. Part (a) [4 points]: Find (a⋅2c)b+||−3c||a. Part (b) [6 points]: Find two perpendicular vectors p and q in R4 such that their sum is the vector b and such that p is parallel to a. Part (c) [3 points]: If T(−1,1,2,−2) is the terminal point of the vector a, then what is its initial point? Part (d) [2 points]: Find a vector in R4 that is perpendicular to b.

Solution :

Given :

a = (1, 2, 3, 4) ,    b = ( 4, 3, 2, 1),    c = (1, 1, 1, 1)     ∈  

a). (a.2c)b + ||-3c||a

Now,

(a.2c) = (1, 2, 3, 4). 2 (1, 1, 1, 1)

         = (2 + 4 + 6 + 6)

         = 20

-3c = -3 (1, 1, 1, 1)

     = (-3, -3, -3, -3)

||-3c|| =

        = 6

Therefore,

(a.2c)b + ||-3c||a = (20)(4, 3, 2, 1) + 6(1, 2, 3, 4)  

                          = (80, 60, 40, 20) + (6, 12, 18, 24)

                         = (86, 72, 58, 44)

b). two vectors and are parallel to each other if they are scalar multiple of each other.

i.e.,   for the same scalar r.

Given is parallel to , for the same scalar r, we have

  ......(1)

Let   ......(2)

Now given  and   are perpendicular vectors, that is dot product of  and   is zero.

 .......(3)

Also given the sum of  and   is equal to . So

∴   ....(4)

Putting the values of in (3),we get

So putting this value of r in (4), we get

These two vectors are perpendicular and satisfies the given condition.

c). Given terminal point is is (-1, 1, 2, -2)

We know that,

Position vector = terminal point - initial point

Initial point = terminal point - position point

                  = (-1, 1, 2, -2) - (1, 2, 3, 4)

                  = (-2, -1, -1, -6)

d).

Let us say a vector  is perpendicular to

Then,

There are infinitely many vectors which satisfies this condition.

Let us choose arbitrary

Therefore,

                      = -3

The vector is (-1, 1, 2, -3) perpendicular to given

d

step-by-step explanation: