Vector Calculus Answers

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.

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°.

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)

find the direction in which the function f=x^2-y^2+xy decreases most rapidly at the point(1,1).

a ⃗,b ⃗,c ⃗are non-zero vectors. If a ⃗×b ⃗= a ⃗×c ⃗ and a ⃗ .b ⃗= a ⃗ .c ⃗ then show that b ⃗= c ⃗.

Prove that|(a ⃗+(b)) ⃗×(a ⃗-(b)) ⃗ | = 2ab. If a ⃗⊥ b ⃗.

Prove that (a ⃗+3b ⃗)×(a ⃗+b ⃗ )+(3a ⃗-5b ⃗)×(a ⃗-b ⃗) = 0.

Show that (a ⃗-d ⃗)×(b ⃗-c ⃗)+(b ⃗-d ⃗)×(c ⃗-a ⃗)+(c ⃗-d ⃗)×(a ⃗-b ⃗) is independent of d ⃗.

If a ⃗=(0,1 ,-1) and c ⃗= (1, 1, 1) are given vectors, then find a vector b ⃗ satisfying a ⃗×b ⃗+c ⃗ = 0 and a ⃗ .b ⃗ = 3.

Let a ⃗ = (3, -1, 0) and b ⃗=(1/2,3/2,1). Find the vector c ⃗satisfying a ⃗×c ⃗=4b ⃗ and a ⃗ .( c) ⃗=1.