Resolution of Vectors, Unit Vectors, Direction Cosines & Vector Geometry - Worksheet

JEE Advanced Worksheet – 2

Higher-Difficulty Single-Concept MCQs

Topic: Resolution of Vectors, Unit Vectors, Direction Cosines & Vector Geometry

Difficulty: JEE Advanced (Single Concept)

Instructions

• 20 Multiple Choice Questions
• Exactly one correct answer
• No calculator
• Assume Cartesian coordinate system.

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Q1. Direction Cosines

A vector has direction ratios $2,-3,6$. Its direction cosine along the z-axis is

A) $\frac67$

B) $\frac37$

C) $\frac23$

D) $\frac6{\sqrt{41}}$

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Q2. The vector

$$\vec A=4\hat i-2\hat j+4\hat k$$

is multiplied by a scalar so that its magnitude becomes 18. The scalar is

A) 2

B) 3

C) $\dfrac32$

D) $\dfrac95$

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Q3.

The unit vector along

$$3\hat i+4\hat j+12\hat k$$

is

A)

$$\frac3{13}\hat i+\frac4{13}\hat j+\frac{12}{13}\hat k$$

B)

$$\frac3{12}\hat i+\frac4{12}\hat j+\hat k$$

C)

$$\frac1{13}(3\hat i+4\hat j+13\hat k)$$

D)

$$\frac3{12}\hat i+\frac4{12}\hat j+\frac{12}{12}\hat k$$

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Q4.

If

$$\vec A+\vec B=\vec0$$

then

A) Both vectors have equal magnitude only

B) Both vectors are perpendicular

C) Both vectors are equal in magnitude and opposite in direction

D) Both are unit vectors

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Q5. If

$$\vec A=2\hat i+3\hat j-\hat k$$

then

$$2(\vec A)-3\hat i+\hat j+5\hat k$$

equals

A) 

$$\hat i+7\hat j+3\hat k$$

B)

$$\hat i+5\hat j+3\hat k$$

C)

$$\hat i+7\hat j+\hat k$$

D)

$$5\hat i+7\hat j+3\hat k$$

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Q6. A vector has magnitude 10 and direction cosines

$$\left(\frac35,\frac45,0\right)$$

Its Cartesian form is

A)

$$6\hat i+8\hat j$$

B)

$$8\hat i+6\hat j$$

C)

$$6\hat i+8\hat j+10\hat k$$

D)

$$3\hat i+4\hat j$$

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Q7. The direction cosines of a vector satisfy

$$l=m=n$$

The angle made with each axis is

A) $30^\circ$

B) $45^\circ$

C) $54.7^\circ$

D) $60^\circ$

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Q8. The magnitude of

$$2\hat i-\hat j+2\hat k$$

is

A) 2

B) 3

C) $\sqrt8$

D) 5

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Q9.

Which one is NOT possible?

A)

$$l^2+m^2+n^2=1$$

B)

$$l=m=n=\frac1{\sqrt3}$$

C)

$$l=1,\ m=1,\ n=0$$

D)

$$l=0,\ m=0,\ n=1$$

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Q10.

A vector is parallel to

$$4\hat i-6\hat j+8\hat k$$

Its unit vector is

A)

$$\frac1{\sqrt{29}}(2\hat i-3\hat j+4\hat k)$$

B)

$$\frac1{\sqrt{116}}(4\hat i-6\hat j+8\hat k)$$

C) Both A and B

D) None

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Q11.

If

$$\vec A=\hat i+\hat j+\hat k$$

then

$$|\vec A+\vec A+\vec A|$$

is

A) 3

B) $3\sqrt3$

C) $9$

D) $9\sqrt3$

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Q12.

The vector joining

$$(2,-1,4)$$

to

$$(5,3,-2)$$

is

A)

$$3\hat i+4\hat j-6\hat k$$

B)

$$-3\hat i-4\hat j+6\hat k$$

C)

$$7\hat i+2\hat j+2\hat k$$

D)

$$3\hat i+2\hat j-6\hat k$$

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Q13.

A particle moves successively by

$$3\hat i,\quad 4\hat j,\quad -3\hat i,\quad 5\hat k$$

Its resultant displacement is

A)

$$4\hat j+5\hat k$$

B)

$$3\hat i+9\hat j$$

C)

$$5\hat k$$

D)

$$4\hat i+5\hat k$$

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Q14.

The magnitude of

$$5(\hat i-\hat j+\hat k)$$

is

A) 5

B) $5\sqrt{\mathrm{2}}$

C) $5\sqrt3$

D) 15

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Q15.If

$$\vec A+\vec B=\vec C$$

and

$$\vec A,\vec B$$

are interchanged, then

A) Resultant changes

B) Magnitude changes

C) Direction changes

D) Resultant remains unchanged

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Q16. A vector has equal projections on x, y and z axes.

Its direction cosine along x-axis is

A)

$$\frac1{\sqrt2}$$

B)

$$\frac1{\sqrt3}$$

C)

$$\frac13$$

D)

$$\frac12$$

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Q17.

If

$$\vec A=2\hat i+\hat j$$

and

$$\vec B=\hat i-4\hat j$$

then

$$3\vec A-2\vec B$$

equals

A)

$$4\hat i+11\hat j$$

B)

$$4\hat i-5\hat j$$

C)

$$8\hat i+11\hat j$$

D)

$$8\hat i-5\hat j$$

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Q18. The magnitude of

$$\hat i+\hat j+\hat k$$

divided by the magnitude of

$$2\hat i+2\hat j+2\hat k$$

is

A)

$$\frac12$$

B)

$$2$$

C)

$$\sqrt2$$

D)

$$\frac1{\sqrt2}$$

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Q19.

The direction ratios proportional to

$$-6,\ 8,\ -2$$

are

A)

$$3,-4,1$$

B)

$$-3,4,-1$$

C)

$$6,8,2$$

D)

$$-6,-8,-2$$

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Q20.

A vector has magnitude 15 and direction cosines

$$\left(\frac23,\frac13,\frac23\right)$$

Its Cartesian form is

A)

$$10\hat i+5\hat j+10\hat k$$

B)

$$15\hat i+10\hat j+15\hat k$$

C)

$$5\hat i+10\hat j+10\hat k$$

D)

$$10\hat i+10\hat j+5\hat k$$

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Answer Key

Q	Ans
1	D
2	C
3	A
4	C
5	A
6	A
7	C
8	B
9	C
10	C
11	B
12	A
13	A
14	C
15	D
16	B
17	A
18	A
19	B
20	A

JEE Advanced concepts covered

• Direction ratios and direction cosines
• Normalization and unit vectors
• Vector geometry in 3D
• Position vectors between two points
• Successive vector operations
• Scalar multiplication and magnitude changes
• Resultant displacement
• Parallel and antiparallel vectors
• Conceptual traps involving commutativity, scaling, and normalization