Vector Calculus Answers

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?

1 If
ϕ=2xz4−x2y
, find
|▽ϕ|
(√93)
(√80)
(√12)
(√110)

2 If
ϕ(x,y,z)=3x2y−y3z2
, find
▽ϕ
at point (1,-2,-1)
−12i−9j−16k
i−3j−k
2i−5j−6k
−3i−4j−2k

3 Find a unit normal to the surface
x2y+2xz=4
at point (2,-2,3)
23i−23j−23k
−15i+25j+25k
−13i+23j+23k
−17i+27j+27k

4 Let
ϕ(x,y,z)=xy2z
and
A=xzi−xy2j+yz2k
,find
∂3∂x2∂z(ϕA)
2i+2j−5k
5i−k
4i−2j
i+j

5 Given that
ϕ=2x2y−xz3
find
▽2ϕ
2y−6xz
4y−6xz
2y−xz
y+6xz

6 If
A=xz3i−2x2yzj+2yz4
, find
▽×A
at point (1,-1,1).
2j+3k
2i+j74k
i+3j+5k
3j+4k

7 Given that
A=A1i+A2j+A3k
and
r=xi+yj+zk
, evaluate
▽⋅(A×r)
if
▽×A=0
0
3
2
5

8 Let
A=x2yi−2xzj+2yzk
, find Curl curl A.
3j+4k
2x+2)k
(2x+2)j
3j−4k

9 Given
A=2x2i−3yzj+xz2k
and
ϕ=2z−x3y
, find
A⋅▽ϕ
at point (1,-1,1).
5
3
4
1

10 Find the directional derivative of
ϕ=x2yz+4xz2
at (1,-2,-1) in the direction
2i−j−2k
373
353
253
113

1 Find the angle between
A=2x+2j−k
and
B=6i−3j+2k
600
450
690
790

2 Determine the value of a so that
A=2i+aj+k
and
B=4i−2j−2k
are perpendicular
a=5
a=3
a=1
a=7

3 Determine a unit vector perpendicular to the plane of
A=2i−6j−3k
and
B=4i+3j−k
±(37i−27j+67k)
±(35i+25j−65k)
±(14i−34j−12k)
±(−23i−13j+34k)

4 Find the work done in moving an object along a vector
r=3i+2j−5k
3
5
7
9

5 Given that
A=2i−j+3k
and
B=3i+2j−k
, find
A⋅B
3
6
1
9

6 If
A=2i−3j−k
and
B=i+4j−2k
, find
(A+B+×(A−B)
3i+4j+25k
2i+6j+2k
−20i−6j−22k
−3i−5j−25k

7 If
A=3i−j+2k
,
B=2i+j−k
and
C=i−2j+2k
, find
(A×B)×C
15i+15j−5k
5i+5j−5k
−10i+10j−5k
15i+10j−5k

8 Determine a unit vector perpendicular to the plane of
A=2i−6j−3k
and
B=4i+3j−k
35i−25j+65
17i−37j+47
37i−27j+67
27i−47j+57

9 Evaluate
(2i−3j)⋅[(i+j−k)×(3i−k)]
4
5
6
8

10 If
A=i−2j−3k
,
B=2i+j−k
and
C=i+3j−2k
,evaluate
(A×B)⋅C
-25
11
15
-20

Determine whether the vectors u and v are parallel, orthogonal, or neither.

u = <6, -2>, v = <8, 24>

a. Neither
b. Parallel
c. Orthogonal

If
A=x^z^3i−2^x^2yzj+2y^z^4
, find
▽×A
at point (1,-1,1).

2j+3k
\
2i+j74k

i+3j+5k

3j+4k

Let
A=2x^2i−3yzj+xz^2k
and
ϕ=2z−x^3y
, find
A.▽ϕ
at point (1,-1,1).
5
3
4
1