Engineering mathematics I

  1. Find the projection of = 3i+2j-4k on = 7i+3j+2k.
  2. Find the projection of on given = 6i+3j-2k and = 2i+4j+2k.
  3. Find the cosine of the angle between the vector =i+5j-3k and = 2i+3j-2k.
  4. Find the sine of the angle between the vectors.
  5. Show that the vectors 4i+5j-k and 2i-3j-7k are orthogonal.
  6. Find the value of ‘m’ if the vectors 2i-5j+mk and 6i+3j+k are orthogonal.
  7. Find the area of a triangle whose sides are 3i+2j-4k and i-2j+5k.
  8. Find the area of parallelogram whose adjacent sides are 4i+3j+k and 2i-j+2k.
  9. Find the unit vector perpendicular to both vectors = 2i-5j+k and =5i+j+7k.
  10. If a force 4i+6j+2k acting on a body displaces it from (2,7,-8) to (3,9,4) find the work done.
  11. A particle is acted by constant forces 3i-j+2k , -i+3j+k and i+j-2k is displaced from point

(-1, 2, 3,) to (2,-1, 5). Calculate the total work done by the forces.

  1. Find the magnitude of moment about a point (1,2,-1) of a force represented by the vector

(3, 0, 1) and acting through the point (2,-1, 3).

  1. If =i+2j-3k &= 3i-5j+2k find the magnitude of 2+3.
  2. If A = (3,-4) & B= (-5, 6) find the position vectors of A & B. Also find ||.
  3. Find the unit vector in the direction of . Given = 2i+3j-4k.
  4. Show that the points with positionvectors -2i+3j-5k, i+2j+3k & 7i-k are collinear.
  5. Show that the points whose position vectors are i-3j-5k, 2i-j+5k & 3i-4j -4k from

Right angled triangle.

  1. Show that position vectors 4i+5j+6k, 5i+6j+4k and 6i +4j +5k from an equilateral triangle.
  2. If position vectors of P, Q, R and S are i+j+k, 2i+5j, 3i+2j-3k and i-6j-k. Show that and

are parallel.

  1. Prove that the vectors and from a triangle.
  2. Express the following Complex number in polar form and exponential form.

1) - -i 2) - +i 3) -1 + i 4) 1 - i 5) 1+ i

  1. Express in form.

1) 3) 4)

  1. Find the Inverse of the matrix

1) 2)

  1. If A = & B = verify = .
  1. If A = & B = P.T adj AB = (adj B) adj A.
  1. If A= show that A.adj A = adjA. A = | A |.
  1. Verify Cayley Hamilton theorem for the matrix

1) 2)

  1. Find the characteristic roots of the matrix

1) 2)

  1. If matrix A= is a Singular matrix find the value of
  1. Evaluate 1)

2)

  1. Find the value of x. If 1) = 0

2) = 3x-1

  1. 32. Solve the following equation by Cramer’srule (or determinant method)

1)x-2y+3z=2, 2x-3z = 3, x+y+z = 6

2)3x+y-z = 4, x+2y+2z = 9, 5x-y+z = 12

  1. IfA = and B = show that (A+B)T = AT+BT.
  1. Find AB given 1). A= and B = .

2). A =.

3).

  1. Find x if