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Vector Calculus
For the vectors □(□(→┬a )) =xyz□(→┬i )-2xz^2 □(→┬j )+xz□(→┬k ) and □(→┬b )=2z□(→┬i )+y□(→┬j )-x□(→┬k)
Find ∂^2/∂x∂y (□(→┬a )×□(→┬b )) at (2,0,-1)
Find ∂^2/∂x∂y (□(→┬a )×□(→┬b )) at (2,0,-1)
Vector Calculus
Find the projection of the line segment joining (2, -1, 3) and (4, 2, 5) on a line which makes equal acute angles with co-ordinate axes.
Vector Calculus
Find the angle between the lines whose direction cosines are given by the equations.
(i) l + 2m +3n = 0 and 3lm – 4ln + mn = 0
(ii) l + m + n = 0 and l2 + m2 – n2 = 0
(i) l + 2m +3n = 0 and 3lm – 4ln + mn = 0
(ii) l + m + n = 0 and l2 + m2 – n2 = 0
Vector Calculus
Prove that in any triangle the line joining the mid-points of any two sides is parallel to the third side and half of its length.
Vector Calculus
The vectors □(→┬a ) and □(→┬b ) are non-collinear. Find for what value of x, the vectors □(→┬c )=(x-2) □(→┬a )+□(→┬b ) and □(→┬d ) =(2x+1) □(→┬a )- □(→┬b ) are collinear.
Vector Calculus
Find the vector method the horizontal force and the force inclined at an angle of 60° to the vertical whose resultant is vertical force p.
Vector Calculus
Find the area of a parallelogram whose diagonals 2□(→┬m-) □(→┬n ) and 4□(→┬m )- 5□(→┬n ) , where □(→┬m ) and □(→┬n ) are unit vectors forming an angle of 45°.
Vector Calculus
1.) For the scalar potential function φ = (x^2 + y^2 + z^2)^2 and the velocity vector field u=(y^2, z, x^2) calculate the following vector quantities:
a.) ∇φ ; ∇⋅ u
b.) (∇^2)φ = (∇⋅ ∇)φ ; (∇^2)u
c.) ∇ × u
where u is a vector, and the vector operator ∇ = ∂/∂x , ∂/∂y, ∂/∂z)
a.) ∇φ ; ∇⋅ u
b.) (∇^2)φ = (∇⋅ ∇)φ ; (∇^2)u
c.) ∇ × u
where u is a vector, and the vector operator ∇ = ∂/∂x , ∂/∂y, ∂/∂z)
Vector Calculus
find the direction in which the function f=x^2-y^2+xy decreases most rapidly at the point(1,1).
Vector Calculus
a ⃗,b ⃗,c ⃗are non-zero vectors. If a ⃗×b ⃗= a ⃗×c ⃗ and a ⃗ .b ⃗= a ⃗ .c ⃗ then show that b ⃗= c ⃗.
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