Vectors and 3D Geometry

21 solved problems from JEE previous year papers.

Vector Algebra

Given a = 4i + 5j − k, b = i − 4j + 5k and c = 3i + j − k, Find a Vector d Which Is Perpendicular to Both b and c and Satisfies d·a = 21

Find a vector d that is perpendicular to both b and c, and satisfies d·a=21, given a=4i+5j-k, b=i-4j+5k, c=3i+j-k. Solved by…
Vector Algebra

Let PQR Be a Triangle With A, B and C on the Sides QR, RP and PQ Such That QA/AR = RB/BP = PC/CQ = 1/2. Then Area(PQR)/Area(ABC) Is Equal To

JEE Main 2023 PYQ: triangle PQR with A,B,C on its sides dividing them in ratio 1:2. Find Area(PQR)/Area(ABC) using position…
Vector Algebra

Using Vectors, Prove That the Line Joining the Midpoints of Two Sides of a Triangle Is Parallel to the Third Side and Exactly Half Its Length

A vector proof of the midpoint theorem: the segment joining the midpoints of two sides of a triangle is parallel to the third…
Vector Algebra

Let the Arc AC of a Circle Subtend a Right Angle at the Centre O. If B on the Arc AC Divides It So That arc AB : arc BC = 1 : 5, and OC = αOA + βOB, Then α + √2(√3−1)β Is Equal To

Arc AC subtends 90° at centre O; point B divides it so arc AB:arc BC=1:5. Given OC=αOA+βOB, find α+√2(√3−1)β by placing the…
Vector Algebra

Prove That for Any Two Vectors a and b, |a + b| ≤ |a| + |b|

Prove the vector triangle inequality |a+b|≤|a|+|b| using the dot product expansion of |a+b|² and the Cauchy-Schwarz bound…
Vector Algebra

If x, y and z Are Three Unit Vectors in Three-Dimensional Space, Then the Minimum Value of |x+y|² + |y+z|² + |z+x|² Is

Given x, y, z are unit vectors in 3D space, find the minimum value of |x+y|²+|y+z|²+|z+x|² by relating the sum to |x+y+z|² and…
Vector Algebra

If a and b Are Unit Vectors, Then the Greatest Value of √3|a+b| + |a−b| Is

Find the greatest value of √3|a+b|+|a−b| where a and b are unit vectors, by writing both magnitudes in terms of the half-angle…
Vector Algebra

If the Vectors a = i − j + 2k, b = 2i + 4j + k and c = λi + j + μk Are Mutually Orthogonal, Then (λ, μ) =

Given a=i−j+2k, b=2i+4j+k, and c=λi+j+μk are mutually orthogonal, find (λ,μ) by setting the dot products a·c and b·c to zero.
Vector Algebra

For p > 0, the Vector v₂ = 2i + (p+1)j Is Obtained by Rotating v₁ = √3 p i + j by an Angle θ About the Origin in a Counterclockwise Direction. If tan θ = (α√3 − 2)/(4√3 + 3), Then the Value of α Is Equal To

For p>0, vector v2=2i+(p+1)j is obtained by rotating v1=√3pi+j by angle θ. Given tanθ=(α√3−2)/(4√3+3), find α using the…
CBSE Delhi 2014

Show That the Lines (x+1)/3 = (y+3)/5 = (z+5)/7 and (x−2)/1 = (y−4)/3 = (z−6)/5 Intersect

Prove that the lines (x+1)/3=(y+3)/5=(z+5)/7 and (x−2)/1=(y−4)/3=(z−6)/5 intersect, by finding parameter values that satisfy all…
AIEEE 2012

Let ABCD Be a Parallelogram Such That AB = q, AD = p and ∠BAD Is an Acute Angle. If r Is the Vector That Coincides With the Altitude Directed From Vertex B to Side AD, Then r Is Given By

In parallelogram ABCD with AB=q, AD=p, and acute ∠BAD, find the vector r for the altitude from vertex B to side AD, using vector…
Vector Algebra

If a, b, c Are Three Mutually Perpendicular Vectors of the Same Magnitude, Prove That a + b + c Is Equally Inclined With a, b and c

Prove that a+b+c makes equal angles with each of a, b, c when a, b, c are three mutually perpendicular vectors of the same…
Three Dimensional Geometry

Let a Straight Line L Pass Through P(2, −1, 3) and Be Perpendicular to the Lines (x−1)/2 = (y+1)/1 = (z−3)/(−2) and (x−3)/1 = (y−2)/3 = (z+2)/4. If L Intersects the yz-Plane at Q, Then the Distance Between P and Q Is

A line L through P(2,−1,3) is perpendicular to two given lines. Find where L meets the yz-plane and the distance PQ, using the…
Three Dimensional Geometry

If the Length of the Perpendicular From the Point (β, 0, β), β ≠ 0, to the Line x/1 = (y−1)/0 = (z+1)/(−1) Is √(3/2), Then β Is Equal To

The perpendicular distance from (β,0,β) to the line x/1=(y−1)/0=(z+1)/(−1) is √(3/2). Find β using the cross-product distance…
Vector Algebra

If A ≡ 2i + 3j, B ≡ pi + 9j and C ≡ i − j Are Collinear, Then the Value of p Is

Points A=2i+3j, B=pi+9j, C=i−j are collinear. Find p by setting the vectors AB and AC parallel using the 2D cross-product…
Vectors

Triangle PQR, QA/AR=RB/BP=PC/CQ=1/2 — Area(PQR)/Area(ABC)

In triangle PQR, points A, B, C lie on QR, RP, PQ with QA/AR = RB/BP = PC/CQ = 1/2. Find the ratio of the area of triangle PQR to…
Vector Algebra

Let a = i + 4j + 2k, b = 3i − 2j + 7k and c = 2i − j + 4k. Find a Vector d Which Is Perpendicular to Both a and b, and Such That c·d = 15

Given a=i+4j+2k, b=3i−2j+7k, c=2i−j+4k, find vector d perpendicular to both a and b with c·d=15, using the cross product a×b.
Three Dimensional Geometry

Direction Cosines of the Line 5x−3=15y+7=3−10z

Find the direction cosines of 5x−3=15y+7=3−10z and a point through which it passes, by converting to symmetric form with…
Vector Algebra

If 3a+2b=5c and 8a−7b=4c, Are a,b,c Collinear and Is |a|>|b|?

Given 3a+2b=5c and 8a−7b=4c, determine whether |a|>|b| and whether a, b, c are collinear vectors, by eliminating c between the…
Vector Algebra

If Four Distinct Points With Position Vectors a, b, c, d Are Coplanar, Then Which of the Following Is True

If four distinct points with position vectors a,b,c,d are coplanar, which scalar triple product identity must hold? A worked…
Vector Algebra

Find Vector d Perpendicular to Both b and c, With d·a = 21

Given vectors a=4i+5j−k, b=i−4j+5k, c=3i+j−k, find vector d perpendicular to both b and c such that d·a=21, by solving a system…