Question 1

If ((â × b̂) × (ĉ × d̂)) · (â × d̂) = 0, which statement must always hold?

Accepted Answer

Either b̂ or ĉ lies in the plane formed by â and d̂.. (a×b)×(c×d) = [a b d]c - [a b c]d. Dotting with (a×d) and using d.(a×d)=0 gives [a b d][a d c] = 0. Hence either [a b d]=0 (b in plane of a,d) or [a d c]=0 (c in plane of a,d): either b or c lies in the plane of a and d.

Question 2

For four points A, B, C and D in space, if the dot product of vectors AB and CD is given by k times [|AD|² + |BC|² - |AC|² - |BD|²], then what is the value of k?

Accepted Answer

1/2. The dot product of vectors AB and CD can be expressed in terms of the lengths of the segments formed by the points A, B, C, and D. The relationship given simplifies to show that the coefficient k must be 1/2 to satisfy the equation, indicating that the geometric configuration of the points leads to this specific proportionality.

Question 3

Three vectors →a, →b and →c have magnitudes |→a| = 4, |→b| = 4 and |→c| = 2. If →a is orthogonal to (→b + →c), →b is orthogonal to (→c + →a), and →c is orthogonal to (→a + →b), then the magnitude of (→a + →b + →c) is:

Accepted Answer

6. Adding a.(b+c)=0, b.(c+a)=0, c.(a+b)=0 gives 2(a.b+b.c+c.a)=0. Then |a+b+c|^2 = 16+16+4+0 = 36, so the magnitude is 6.

Question 4

The position vectors of the vertices A, B and C of triangle ABC are 4î + 7ĵ + 8k̂, 2î + 3ĵ + 4k̂ and 2î + 5ĵ + 7k̂ respectively. The position vector of the point where the internal bisector of angle A intersects BC is

Accepted Answer

(1/3)(6î + 13ĵ + 18k̂). AB=6, AC=3, so the bisector from A meets BC at D dividing it in ratio AB:AC=2:1 (BD:DC). D=(2*C+1*B)/3=(2,13/3,6)=(1/3)(6i+13j+18k).

Question 5

Let a=a₁î+a₂ĵ+a₃k̂, b=b₁î+b₂ĵ+b₃k̂, and c=c₁î+c₂ĵ+c₃k̂ be three non-zero vectors. Suppose c is a unit vector perpendicular to both a and b. If the angle between a and b is π/6, then the value of |[a₁, a₂, a₃; b₁, b₂, b₃; c₁, c₂, c₃] | is

Accepted Answer

(1)/(4)|a|²|b|². c is a unit vector along a x b, so [a b c] = c.(a x b) = |a x b| = |a||b| sin(pi/6) = |a||b|/2. The squared value (as the options are framed) is (1/4)|a|^2|b|^2.

Question 6

Let a, b, and c be three vectors that are not in the same plane. Define p = (b × c)/([abc]), q = (c × a)/([abc]), and r = (a × b)/([abc]). Then the value of (a+b)·p + (b+c)·q + (c+a)·r is

Accepted Answer

3. p,q,r form the reciprocal basis, so a.p=b.q=c.r=1 and all cross terms (b.p, c.q, a.r, etc.) = 0. Thus (a+b).p = 1, (b+c).q = 1, (c+a).r = 1, giving total = 3.

Question 7

A vector a = α î + 2ĵ + β k̂ lies in the plane determined by b = î + ĵ and c = ĵ + k̂, and it bisects the angle between b and c. Which of the following can be the values of α and β?

Accepted Answer

α = 1, β = 1. Since |b|=|c|=sqrt2, the internal bisector is along b+c=(1,2,1). For a=(alpha,2,beta) to be parallel to (1,2,1) with middle component 2, take the vector (1,2,1) itself, giving alpha=1, beta=1.

Question 8

Let p, q, r be three pairwise perpendicular vectors, each having the same magnitude. If a vector x satisfies

p×[(x-q)×p]+q×[(x-r)×q]+r×[(x-p)×r]=0,

then x is

Accepted Answer

(1)/(2)(p+q+r). With p,q,r mutually perpendicular and equal magnitude, each triple product term p x [(x-q) x p] = |p|^2(x-q)-(p.(x-q))p. Summing all three and setting equal to 0 yields x = (1/2)(p+q+r).

Question 9

Three non-coplanar unit vectors a, b, and c are pairwise inclined at the same acute angle θ, and they are not collinear. The value of |a |b |c| in terms of θ is

Accepted Answer

(1-cosθ)√(1+2cosθ). |[a b c]|^2 equals the Gram determinant with 1 on the diagonal and cos(theta) off-diagonal = 1 - 3cos^2(theta) + 2cos^3(theta) = (1-cos theta)^2(1+2cos theta). So |[a b c]| = (1-cos theta)*sqrt(1+2cos theta).

Question 10

Three forces, hat i+2hat j-3hat k, 2hat i+3hat j+4hat k, and -hat i-hat j+hat k, act at the point (0,1,2). The resultant moment of these forces about the point A(1,-2,0) is

Accepted Answer

-2hat i+6hat j-10hat k. The correct option is derived from calculating the moment generated by each force about point A by using the cross product of the position vector from A to the point of application of the force and the force vector itself. Summing these moments gives the resultant moment, which matches the provided answer.

Question 11

If a and b are not parallel, then for what value of α will the vectors u=(α-2)a+b and v=(2+3α)a-3b be collinear?

Accepted Answer

2/3. For collinearity u=lambda*v: (alpha-2)=lambda(2+3alpha) and 1=-3lambda so lambda=-1/3. Substituting gives alpha-2=-(2+3alpha)/3, leading to 2alpha=4/3, alpha=2/3.

Question 12

Given vectors a=î+ĵ and b=2ĵ-k̂, a vector r satisfies r×a=b×a and r×b=a×b. Find (r)/(|r|).

Accepted Answer

(î+3ĵ-k̂)/(√(11)). r x a = b x a => (r-b) x a = 0 => r = b + s a; r x b = a x b => r = a + t b. Consistency gives r = a + b = (1,3,-1). So r/|r| = (i+3j-k)/sqrt(11).

Question 13

In triangle ABC, the points D, E and F lie on BC, CA and AB respectively such that BD:DC = 1:4, CE:EA = 3:2 and AF:FB = 3:7. Also, K is a point on AB with AK:KB = 1:3. Then the ratio (AD+BE+CF): CK is

Accepted Answer

2: 5. Using position vectors, AD+BE+CF = (3A+B-4C)/10 while CK = (3A+B-4C)/4. They are parallel, so the magnitude ratio is (1/10):(1/4) = 4:10 = 2:5.

Question 14

A parallelogram has adjacent sides given by the vectors (1)/(√2)î+(1)/(√2)ĵ+k̂ and (1)/(√2)î-(1)/(√2)ĵ+k̂. What is the measure of its acute interior angle?

Accepted Answer

60°. With u=(1/sqrt2,1/sqrt2,1) and v=(1/sqrt2,-1/sqrt2,1), u.v = 1/2-1/2+1 = 1 and |u|=|v|=sqrt2. So cos(angle)=1/2, giving 60 degrees, which is the acute interior angle.

Question 15

Three vectors a, b, c satisfy a≠ 0 and a×b=2 a×c. Also, |a|=|c|=1, |b|=4, and the angle between b and c is cos⁻¹ ((1)/(4)). If b-2c=λ a, then the value of λ is

Accepted Answer

± 4. The equation ( ext{b} - 2 ext{c} = ext{λa} ext{ implies that the vector ( ext{b} - 2 ext{c} ext{ is parallel to ( ext{a} ext{. Given the magnitudes and the relationship between the vectors, we find that ( ext{λ} ext{ must equal ( ext{±4} ext{ to satisfy the conditions of the problem, particularly the magnitudes and the angle between ( ext{b} ext{ and ( ext{c} ext{.

Question 16

For an arbitrary vector p, what is the value of (3/2) [|p × î|² + |p × ĵ|² + |p × k̂|²] ?

Accepted Answer

3p². |p x i|^2 + |p x j|^2 + |p x k|^2 = (p2^2+p3^2)+(p1^2+p3^2)+(p1^2+p2^2) = 2(p1^2+p2^2+p3^2) = 2p^2. Multiplying by 3/2 gives 3p^2.

Question 17

If the magnitudes of the vectors a + b and a − b are equal, then the vectors a and b represent

Accepted Answer

a rectangle. |a+b|^2=|a-b|^2 gives 4(a.b)=0, so a and b are perpendicular. As adjacent sides of a parallelogram, perpendicular sides form a rectangle (a square would also require equal magnitudes, which is not given).

Question 18

Three vectors aî+ĵ+k̂, î+ĵ+k̂, and î+ĵ+ck̂ are coplanar, where a, b, and c are distinct and none of them is equal to 1. Find the value of (1)/(1-a)+(1)/(1-b)+(1)/(1-c).

Accepted Answer

1. For (a,1,1),(1,b,1),(1,1,c) coplanar, the determinant abc - a - b - c + 2 = 0. Dividing through leads to the identity 1/(1-a) + 1/(1-b) + 1/(1-c) = 1.

Question 19

Consider the following statements:

Statement 1: If vectors a = 2î + k̂, b = 3ĵ + 4k̂, and c = 8î − 3ĵ are coplanar, then c = 4a − b.

Statement 2: A collection of vectors a1, a2, a3,..., an is called linearly independent if the equation ℓ1a1 + ℓ2a2 +... + ℓnan = 0 can hold only when ℓ

Accepted Answer

Statement-1 is True, Statement-2 is True: Statement-2 is NOT a correct explanation for Statement-1. Statement-1 is true because the given vectors can be expressed in terms of each other, confirming their coplanarity. Statement-2 accurately defines linear independence, but it does not directly explain the coplanarity condition of the vectors in Statement-1.

Question 20

Let A = 2î + 3ĵ + 6k̂, B = î + ĵ - 2k̂, and C = î + 2ĵ + k̂. Find the value of |A × [A × (A × B) ] · C |.

Accepted Answer

343. The expression involves multiple vector cross products and the dot product, ultimately simplifying to a scalar value. The correct option is derived from the properties of vector operations, specifically the triple product and the magnitudes involved, leading to the final result of 343.

Question 21

In triangle ABC, the side vectors are AB = 3i + 4k and AC = 5i − 2j + 4k. What is the magnitude of the median drawn from vertex A?

Accepted Answer

√33. The median from vertex A to side BC can be calculated using the formula for the length of a median in a triangle, which involves the lengths of the sides and the coordinates of the vertices. By finding the midpoint of side BC and then calculating the distance from vertex A to this midpoint, we find that the magnitude of the median is √33.

Question 22

Three vectors a, b, and c satisfy a + b + c = 0. If |a| = 1, |b| = 2, and |c| = 3, then the value of a·b + b·c + c·a is

Accepted Answer

−7. The equation a + b + c = 0 implies that c = -a - b. By substituting this into the expression a·b + b·c + c·a and using the magnitudes of the vectors, we can derive that the result simplifies to -7, confirming the correct answer.

Question 23

If u, v, and w are three vectors that are not in the same plane, then the value of (u+v-w)· ((u-v)×(v-w)) is

Accepted Answer

u·(v× w). The expression (u+v-w) represents a linear combination of the vectors, and when dotted with the cross product ((u-v) imes(v-w)), it simplifies to the scalar triple product, which is equal to uullet(v imes w). This is because the scalar triple product gives the volume of the parallelepiped formed by the three vectors, and since they are not coplanar, this value is non-zero.

Question 24

Let a, b and c be three non-zero vectors such that no two of these are collinear. If the vector a + 2b is collinear with c and b + 3c is collinear with λ (λ being some non-zero scalar) then a + 2b + 6c equals

Accepted Answer

0. Since a+2b is parallel to c, a+2b+6c=(a+2b)+6c is a multiple of c. Since b+3c is parallel to a, a+2b+6c=a+2(b+3c) is a multiple of a. As a and c are non-collinear, the common vector must be 0, so a+2b+6c=0.

Question 25

If a, b, c are non-coplanar vectors and λ is a real number, then the vectors a + 2b + 3c, λb + 4c and (2λ - 1)c are non coplanar for

Accepted Answer

all except two values of λ. The three vectors are non-coplanar iff the determinant of their coefficient matrix [[1,2,3],[0,lambda,4],[0,0,2lambda-1]] = lambda(2lambda-1) is nonzero, which fails only at lambda=0 and lambda=1/2. So they are non-coplanar for all except two values of lambda.

Question 26

Let u, v, w be such that |u| = 1, |v| = 2, |w| = 3. If the projection of v along u is equal to that of w along u and v, w are perpendicular to each other, then |u - v + w| equals

Accepted Answer

√14. The projection of v along u is given by the formula (v · u)u, and since the projections of v and w along u are equal, we can set up an equation involving their magnitudes. Given that v and w are perpendicular, we can use the Pythagorean theorem to find the magnitude of the vector u - v + w, leading to the result |u - v + w| = √14.

Question 27

Let a, b and c be non-zero vectors such that (a × b) × c = 1/3 |b| |c| a. If q is the acute angle between the vectors b and c, then sin q equals

Accepted Answer

2√2 / 3. Using (a x b) x c = (a.c)b - (b.c)a = (1/3)|b||c|a gives a.c=0 and b.c = -(1/3)|b||c|. For the acute angle q, cos q = 1/3, so sin q = sqrt(1 - 1/9) = 2sqrt(2)/3.

Question 28

A speed of 1/4 m/s is split into two components along OA and OB, where OA and OB make angles of 30° and 45°, respectively, with the given velocity. The magnitude of the component along OB is

Accepted Answer

1/8 (√6 − √2) m/s. The correct option is derived from resolving the velocity vector into its components along OA and OB using trigonometric functions. By applying the sine of the angle for OB, we find that the magnitude of the component along OB is indeed 1/8 (√6 − √2) m/s, which accurately represents the projection of the original speed in that direction.

Question 29

If C is the midpoint of the line segment AB and P is any point not lying on AB, then which relation is true?

Accepted Answer

PA + PB = 2PC. The correct option states that the sum of the vectors from point P to points A and B equals twice the vector from P to the midpoint C. This relationship holds true because the midpoint C divides the segment AB into two equal parts, making the distances PA and PB symmetrical around C.

Question 30

For an arbitrary vector a, the quantity (a × î)² + (a × ĵ)² + (a × k̂)² is equal to

Accepted Answer

2a². The expression represents the sum of the squares of the magnitudes of the cross products of vector a with the unit vectors i, j, and k. Each term contributes a factor of a², and due to the properties of cross products, the total simplifies to 2a².

Question 31

If a, b, c are non-coplanar vectors and λ is a real number, then [λ(a+b), λ²b, λc] = [a, b+c, b] holds for

Accepted Answer

no real value of λ. The equation involves determinants of vectors, and since a, b, and c are non-coplanar, their determinant is non-zero. The left-hand side scales the vectors by powers of lambda, which cannot equal the linear combination on the right-hand side for any real lambda, indicating that there are no real solutions.

Question 32

Let a = î - k̂, b = xî + ĵ + (1-x)k̂ and c = yî + xĵ + (1+x-y)k̂. Then [a, b, c] depends on

Accepted Answer

neither x nor y. The scalar triple product [ a, b, c] is determined by the vectors' linear independence and their orientation in space, which in this case does not change with variations in x or y, indicating that the result is independent of both variables.

Question 33

ABC is a triangle. Forces P, Q, R acting along IA, IB, and IC respectively are in equilibrium, where I is the incentre of △ ABC. Then P: Q: R is

Accepted Answer

cos (A)/(2): cos (B)/(2): cos (C)/(2). For forces along IA, IB, IC at the incentre I in equilibrium, each force is proportional to the cosine of half the opposite... actually each is proportional to cos of half its own vertex angle: P:Q:R = cos(A/2):cos(B/2):cos(C/2).

Question 34

The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is

Accepted Answer

3: 2√(2). The problem involves two forces, where the resultant is perpendicular to one of them, indicating a right triangle relationship. By applying the Pythagorean theorem and the given conditions, we can derive the ratio of the larger force to the smaller force, which simplifies to 3: 2√2.

Question 35

ABC is a triangle, right angled at A. The resultant of the forces acting along AB, BC with magnitudes (1)/(AB) and (1)/(AC) respectively is the force along AD, where D is the foot of the perpendicular from A on to BC. The magnitude of the resultant is

Accepted Answer

(1)/(AD). Forces 1/AB and 1/AC act along the perpendicular sides AB and AC, so the resultant magnitude is sqrt((1/AB)^2 + (1/AC)^2). Since AD is the altitude to the hypotenuse, 1/AD^2 = 1/AB^2 + 1/AC^2, hence the magnitude equals 1/AD.

Question 36

If (a×b)×c = a×(b×c) where a, b and c are any three vectors such that a·b ≠ 0, b·c ≠ 0 then a and c are

Accepted Answer

parallel. (a x b) x c = (a.c)b - (b.c)a and a x (b x c) = (a.c)b - (a.b)c. Equality gives (b.c)a = (a.b)c; since a.b != 0 and b.c != 0, a and c must be parallel.

Question 37

The values of a, for which points A, B, C with position vectors 2î-ĵ+k̂, î-3ĵ-5k̂ and aî-3ĵ+k̂ respectively are the vertices of a right angled triangle with C=π/2 are

Accepted Answer

2 and 1. CA = A-C = (2-a, 2, 0) and CB = B-C = (1-a, 0, -6). For angle C = 90 deg, CA.CB = (2-a)(1-a) = 0, so a = 2 or a = 1. Answer: 2 and 1.

Question 38

A particle has two velocities of equal magnitude inclined to each other at an angle θ. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant velocity. Then θ is

Accepted Answer

120°. Let both speeds = v at angle theta; the original resultant bisects theta. After halving one, new resultant R'=A+B/2 makes angle theta/4 with A, so tan(theta/4)=(sin theta)/(2+cos theta). Testing theta=120: RHS=(sqrt3/2)/(2-1/2)=0.577=tan30=tan(120/4). So theta=120 deg.

Question 39

The non-zero vectors a, b, and c satisfy a=8b and c=-7b. What is the angle between a and c?

Accepted Answer

π. The vectors a and c are scalar multiples of the same vector b, with a pointing in the same direction as b and c pointing in the opposite direction. This means that a and c are directly opposite to each other, resulting in an angle of pi (180 degrees) between them.

Question 40

A vector has projections 6, -3, and 2 on the x-, y-, and z-axes respectively. Its direction cosines are:

Accepted Answer

(6)/(7),(-3)/(7),(2)/(7). The correct option represents the direction cosines of the vector, which are calculated by dividing each projection by the vector's magnitude. The magnitude is found using the formula √(6² + (-3)² + 2²) = 7, leading to the direction cosines (6)/(7), (-3)/(7), (2)/(7).

Question 41

Given the vectors a=î-ĵ+2k̂, b=2î+4ĵ+k̂, and c=λî+ĵ+μk̂, if all three are pairwise perpendicular, then the pair (λ,μ) is

Accepted Answer

(-3, 2). The vectors are pairwise perpendicular if their dot products are zero. By calculating the dot products of ( ext{a} cdot c, b cdot c, a cdot b, and solving the resulting equations, we find that the values of ( ext{lambda} and ( ext{mu} that satisfy all conditions are (-3, 2).

Question 42

Given a=(1)/(√(10))(3î+k̂) and b=(1)/(7)(2î+3ĵ-6k̂), find the value of (2a-b) [(a×b)×(a+2b) ].

Accepted Answer

-5. The expression involves vector operations where the cross product and dot product are calculated. The correct option is -5 because the resulting vector from the cross product is orthogonal to both vectors involved, and the dot product with the linear combination of vectors yields this specific value.

Question 43

The vectors a and b are not perpendicular and c and d are two vectors satisfying b×c = b×d and a·d = 0. Then the vector d is equal to

Accepted Answer

c - ((a·c)/(a·b)) b. The correct option is derived from the condition that the cross products of b with c and d are equal, indicating that d can be expressed as a linear combination of c and a vector in the direction of b. The term ((a·c)/(a·b)) b adjusts c by projecting it onto the direction of b, ensuring that d remains orthogonal to a, as required by the condition a·d = 0.

Question 44

If the vectors i+pj+k, i+qj+k, and i+j+rk are coplanar, where p, q, and r are distinct and none of them is equal to 1, then the value of pqr-(p+q+r) is

Accepted Answer

-2. The vectors are coplanar if the scalar triple product is zero. By calculating the determinant formed by these vectors, we find that the condition leads to the equation pqr - (p + q + r) = -2, confirming that the correct answer is -2.

Question 45

Let a, b, c be three non-zero vectors, no two of which are collinear. If a + 3b is parallel to c and b + 2c is parallel to a, then the vector a + 3b + 6c is:

Accepted Answer

0. The conditions given imply that the vectors can be expressed in terms of each other, leading to a situation where the combination of these vectors results in the zero vector. Specifically, since both expressions are parallel to different vectors, their linear combinations ultimately cancel each other out, resulting in the vector being zero.

Question 46

Let a and b be two unit vectors. If the vectors c = a + 2b and d = 5a - 4b are perpendicular to each other, then the angle between a and b is:

Accepted Answer

π/3. The vectors c and d being perpendicular means their dot product is zero. By substituting the expressions for c and d and using the properties of unit vectors, we can derive the cosine of the angle between a and b, leading to the conclusion that the angle is π/3.

Question 47

Let ABCD be a parallelogram such that AB = q, AD = p and ∠DAB an acute angle. If r is the vector that coincide with the altitude directed from the vertex B to the side AD, then r is given by:

Accepted Answer

r = -q + ((p·q)/(p·p)) p. The correct option represents the vector from point B to the line AD, calculated by subtracting the projection of vector q onto vector p from vector q itself. This projection accounts for the angle between the vectors, ensuring that the resultant vector r is perpendicular to AD, which is the definition of an altitude.

Question 48

If the vectors AB = 3i + 4k and AC = 5i - 2j + 4k are the sides of a triangle ABC, then the length of the median through A is

Accepted Answer

√33. The length of the median from vertex A to the midpoint of side BC can be calculated using the formula for the median in a triangle, which involves the lengths of the sides. By applying the median formula and simplifying, we find that the length of the median through A is √33.

Question 49

Let a, b and c be three unit vectors such that a×(b×c) = (√3/2)(b + c). If b is not parallel to c, then the angle between a and b is:

Accepted Answer

5π/6. Expanding, a x (b x c) = (a.c)b - (a.b)c = (sqrt(3)/2)(b + c). Since b is not parallel to c, compare coefficients: a.c = sqrt(3)/2 and -(a.b) = sqrt(3)/2, so a.b = -sqrt(3)/2. With unit vectors, cos(theta) = -sqrt(3)/2, giving theta = 5*pi/6.

Question 50

Let a = 2i + j - 2k and b = i + j. Let c be a vector such that |c - a| = 3, |(a×b)×c| = 3 and the angle between c and a×b be 30°. Then a·c is equal to:

Accepted Answer

2. a=(2,1,-2), a x b=(2,-2,1) with |a x b|=3. From |(a x b) x c|=3=3|c|sin30 -> |c|=2. Then |c-a|^2=9 -> |c|^2-2(a.c)+|a|^2=9 -> 4-2(a.c)+9=9 -> a.c=2.

JEE Main Maths: Vector Algebra questions with solutions

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