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If F=A×(B×C) where A=5it^2 + (3t-2)j + 6tk; B=2i+4tj+3(1-t)k and C=4ti+5jt^2 -3tk evaluate integral of Fdt with limits zero to one.
What do you understand by unit tangent vector

Prove that the divergence of a curl is always zero:

∇⃗⃗. (∇⃗⃗

×a)=0

The position vector r of a particle at time t is
r = 2t
2
i + (t
2 − 4t)j + (3t − 5)k.
Find the velocity and the acceleration of the particle at time t. Show that when t =
2
5
the velocity
and the acceleration are perpendicular to each other. The velocity and the acceleration are resolved
into components along and perpendicular to the vector i − 3j + 2k. Find the velocity and acceleration
components parallel to this vector when t =
2
5
1. The position vector r of a missile at any time t is given by
r = ti + (8t − t
2
)j,
where i and j are unit vectors in the horizontal and upward vertical directions. Find its
(a) average velocity from t = 2 to t = 3,
(b) velocity, speed and direction of motion when t = 4,
(c) position vector when it is moving parallel to the vector i − 2j,
(d) acceleration vector.
2. The position vector r of a moving particle at time t after the start of the motion is given by
r = 5(1 + 4t)i + 5(19 + 2t − t
2
)j.
Find the initital velocity of the particle. At time t = T, the particle is moving at right angles to its initial
direction of motion. Find the value of T and the distance of the particle from its initial position at this
time.
Find the laplacian of the function (phi(x,y,z)=2x^2y-xzn^3
For a particle undergoing circular motion with an angular velocity w in in a circle of radius r show that
w×(w×r) = -w^2.r
) For a scalar field ( , , ) ,
n n n
φ x y z = x + y + z show that (∇φ) r = nφ
r r
. , where n is a non-zero
real constant.
5. Given that vectors
a→=5i^−2j^+3k^
b→=3i^+j^−2k^
c→=i^−3j^+4k^
, calculate the scalar triple product
a⃗ ⋅(b⃗ ×c⃗ )

6. Given that vectors
a→=5i^−2j^+3k^,
b→=3i^+j^−2k^
and
c→=i^−3j^+4k^
, calculate the vector triple product
a⃗ ×(b⃗ ×c⃗ )
why there is no any particular rule to subtract vectors?
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