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