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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

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.

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 )

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

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

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

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

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

A=2x^2i−3yzj+xz^2k

and

ϕ=2z−x^3y

, find

A.▽ϕ

at point (1,-1,1).

5

3

4

1

Let

A=x^2yi−2xzj+2yzk

, find Curl curl A.

3j+4k

2x+2)k

(2x+2)j

3j−4k

A=x^2yi−2xzj+2yzk

, find Curl curl A.

3j+4k

2x+2)k

(2x+2)j

3j−4k