Linear Algebra Answers

For any four vectors a^ vector, b ^ vector, C ^ vector and d ^ vector determine : (a^ vector × b^ vector)(c ^ vector × d ^ vector) + (b^ vector × c^ vector) (a^vector × d^vector) + (c^vector × a^ vector) ( b^vector × d^ vector).

Obtain a unit vector perpendicular to the plane of the vectors A^vector = 3i^cap - 4j^cap +2k^cap and B ^vector = i^cap + j ^ cap +3 k^ cap.

Check whether the following system of equations has a solution
x + y + 3z + w = 5
-x + y + z - 5w = 7
x + 2y + 5z - w = 5

Find the radius and the center of the circular section of the sphere |r| = 17 cut off by
the plane r . (i + 2j + 2k) = 24.

Find the equation of the line passing through the point A(1, 0, -1) and parallel to the
line joining B(1, 2, 3) and C( -1, 2, 0). Is it perpendicular to the line
(1 + 3α , 0 ,1 - 2α ).

Let V=R2 . Define addition + on V by ( x1 , y1 ) + ( x2 , y2 ) = ( x1+x2 , y1+y2 ) and scalar multiplication . by r . (a , b) = (ra , 0). Check whether V satisfies all the
conditions for it to be a vector space over R with respect to these operations.

use the Gaussian Elimination method to find the value of a so that the system of equations.
x+(a+4)y+(4a+2) z=0
2x+3ay+(3a+4) z=0
x+2(a+1)y+(3a+4) z=0
has (i) a unique solution ;
(ii) infinitely many solutions.
Further, find the solution set in each case.

If A B C are three vectors show that A*(B+C)=A*B+B*C

If a and b are non-collinear vectors and A = (x+y)a+(2x+y+1)b
A=(x+y)a+(2x+y+1)b

a) x=2,y=4
b) x=2,y=1
c)x=4,y2
d)x=1,y=1