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🚀 COMPLETE PREPARATION LOOP: LEARN → PRACTICE → ANALYZE

Vector Algebra & 3D Geometry Preparation Hub

Master Vector Algebra & 3D Geometry with coaching-grade theory notes, verified video lectures, and 10 interactive MCQs with instant mistake analysis. Continue with 500+ adaptive questions inside the Exam Sprinter app.

📘 Theory Notes ✍️ Interactive Practice ⏱️ CBT Simulator

What is Vector Algebra & 3D Geometry?

Vector Algebra & 3D Geometry covers the essential principles and concepts required for JEE Main.

💡 Why Study Vector Algebra & 3D Geometry?

Highly important, frequently tested in JEE Main.

Parameter	Details / Relevance
Target Exam	JEE Main
Subject Category	Mathematics
Estimated Study Duration	10 Hours
Expected Questions	2 Questions
Difficulty Index	Medium
Interactive Solved MCQs	10 Questions with AI Diagnostics
Adaptive App Practice	500+ Questions & Real-Time AI Tutor

⚠️ Common Pitfalls to Avoid

Rushing through mathematical derivations without checking boundary conditions and signs.
Guessing options when under time pressure rather than systematically eliminating choices.
Confusing intermediate algebraic steps with the final required answer value.

✓ COACHING-GRADE CLASSROOM NOTES

Vector Algebra & 3D Geometry Theory & Derivations

Browse All Notes Library ›

Access comprehensive classroom-grade derivations and NCERT micro-extracts for Vector Algebra & 3D Geometry. Features step-by-step proofs and standard assumptions.

Important Formulas & Cheat Sheet

$$\vec{a}\cdot\vec{b} = |a||b|\cos\theta$$
$$|\vec{a}\times\vec{b}| = |a||b|\sin\theta$$
Dist (point to plane): $$d = \frac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}}$$

✍️ STEP 3: PRACTICE & ANALYZE

Vector Algebra & 3D Geometry Interactive Practice

⏱️ Take Chapter CBT Test

Solve these 10 standard exam-level questions. The system tracks your response time, detects rapid guessing via Cognitive Reading Thresholds, and outputs your post-session diagnosis.

Interactive Practice Progress:0 / 10 Answered

⏱️ Launch Chapter CBT Simulator

Q1Dot & Cross Products

Expected: 60sEasy

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

Q2Triple Products

Expected: 60sEasy

The vector $\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k$ lies in the plane of the vectors $\overrightarrow b = \widehat i + \widehat j$ and $\overrightarrow c = \widehat j + \widehat k$ and bisects the angle between $\overrightarrow b$ and $\overrightarrow c$ .Then which one of the following gives possible values of $\alpha$ and $\beta$ ?

Q3Lines & Planes in 3D Space

Expected: 60sEasy

$\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(a \hat{i}-\hat{j}), a \neq 0$ and $\vec{r}=(4 \hat{i}-\hat{k})+\mu(2 \hat{i}+a \hat{k})$ from the origin is :

Q4Dot & Cross Products

Expected: 60sEasy

Let $\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$ be two vectors. Then which one of the following statements is TRUE ?

Q5Triple Products

Expected: 90sMedium

If the mirror image of the point (2, 4, 7) in the plane 3x $-$ y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to :

Q6Lines & Planes in 3D Space

Expected: 90sMedium

Two lines ${{x - 3} \over 1} = {{y + 1} \over 3} = {{z - 6} \over { - 1}}$ and ${{x + 5} \over 7} = {{y - 2} \over { - 6}} = {{z - 3} \over 4}$ intersect at the point R. The reflection of R in the xy-plane has coordinates :

Q7Dot & Cross Products

Expected: 90sMedium

Let $\vec{a}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}, \vec{b}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\vec{c}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ . Let $\vec{v}$ be the vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ , such that the length of its projection on the vector $\vec{c}$ is $\frac{1}{\sqrt{14}}$ . Then $|\vec{v}|$ is equal to

Q8Triple Products

Expected: 90sMedium

The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and passing through the point (1, 1, 0) is :

Q9Lines & Planes in 3D Space

Expected: 120sHard

Let the lines $l_{1}: \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2}$ and $l_{2}: 3 x+2 y+z-2=0=x-3 y+2 z-13$ be coplanar. If the point $\mathrm{P}(a, b, c)$ on $l_{1}$ is nearest to the point $\mathrm{Q}(-4,-3,2)$ , then $|a|+|b|+|c|$ is equal to

Q10Dot & Cross Products

Expected: 120sHard

Let the equation of plane passing through the line of intersection of the planes $x+2 y+a z=2$ and $x-y+z=3$ be $5 x-11 y+b z=6 a-1$ . For $c \in \mathbb{Z}$ , if the distance of this plane from the point $(a,-c, c)$ is $\frac{2}{\sqrt{a}}$ , then $\frac{a+b}{c}$ is equal to :

🗺️ Recommended Learning Sequence

Coordinate Geometry➔Vector Algebra & 3D Geometry (Current)

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Vector Algebra & 3D Geometry Preparation Hub

What is Vector Algebra & 3D Geometry?

💡 Why Study Vector Algebra & 3D Geometry?

⚠️ Common Pitfalls to Avoid

Vector Algebra & 3D Geometry Theory & Derivations

Important Formulas & Cheat Sheet

Vector Algebra & 3D Geometry Interactive Practice

If the vectors AB→=3i^+4k^\overrightarrow {AB} = 3\widehat i + 4\widehat kAB=3i+4k and AC→=5i^−2j^+4k^\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat kAC=5i−2j​+4k are the sides of a triangle ABC,ABC,ABC, then the length of the median through AAA is :

r⃗=(i^+j^−k^)+λ(ai^−j^),a≠0\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(a \hat{i}-\hat{j}), a \neq 0r=(i^+j^​−k^)+λ(ai^−j^​),a=0 and r⃗=(4i^−k^)+μ(2i^+ak^)\vec{r}=(4 \hat{i}-\hat{k})+\mu(2 \hat{i}+a \hat{k})r=(4i^−k^)+μ(2i^+ak^) from the origin is :

Let a⃗=5i^−j^−3k^\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}a=5i^−j^​−3k^ and b⃗=i^+3j^+5k^\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}b=i^+3j^​+5k^ be two vectors. Then which one of the following statements is TRUE ?

If the mirror image of the point (2, 4, 7) in the plane 3x −-− y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to :

The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and passing through the point (1, 1, 0) is :

🗺️ Recommended Learning Sequence

Ready to achieve complete mastery in Vector Algebra & 3D Geometry?

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

$\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(a \hat{i}-\hat{j}), a \neq 0$ and $\vec{r}=(4 \hat{i}-\hat{k})+\mu(2 \hat{i}+a \hat{k})$ from the origin is :

Let $\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$ be two vectors. Then which one of the following statements is TRUE ?

If the mirror image of the point (2, 4, 7) in the plane 3x $-$ y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to :