📌 1. Introduction to Vectors

A vector is a quantity that has both:

• Magnitude
• Direction

Examples:

• Displacement
• Velocity
• Force

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📊 Representation of a Vector

A vector is represented as:$$\vec{a}$$a

Or in component form:$$\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$$a=a1​i^+a2​j^​+a3​k^

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$\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$a=a1​i^+a2​j^​+a3​k^

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📖 2. Types of Vectors

🔹 Zero Vector

Magnitude = 0

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🔹 Unit Vector

Magnitude = 1$$\hat{a} = \frac{\vec{a}}{|\vec{a}|}$$a^=∣a∣a​

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$\hat{a} = \frac{\vec{a}}{|\vec{a}|}$a^=∣a∣a​

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🔹 Equal Vectors

Same magnitude and direction

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🔹 Collinear Vectors

Parallel vectors

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🔹 Coplanar Vectors

Vectors lying in same plane

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📐 3. Magnitude of a Vector

For vector:$$\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$$a=a1​i^+a2​j^​+a3​k^

Magnitude:$$|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$$∣a∣=a12​+a22​+a32​​

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$|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$∣a∣=a12​+a22​+a32​​

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➕ 4. Addition of Vectors

If:$$\vec{a} = (a_1, a_2, a_3), \quad \vec{b} = (b_1, b_2, b_3)$$a=(a1​,a2​,a3​),b=(b1​,b2​,b3​)

Then:$$\vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)$$a+b=(a1​+b1​,a2​+b2​,a3​+b3​)

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📌 Properties:

• Commutative
• Associative

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➖ 5. Subtraction of Vectors

$$\vec{a} – \vec{b} = \vec{a} + (-\vec{b})$$a−b=a+(−b)

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✖️ 6. Scalar Multiplication

If $k$k is a scalar:$$k\vec{a} = (ka_1, ka_2, ka_3)$$ka=(ka1​,ka2​,ka3​)

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🔗 7. Position Vector

Vector representing position of a point from origin.

Example:
Point $P(x, y, z)$P(x,y,z)$$\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$$OP=xi^+yj^​+zk^

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📏 8. Section Formula

If point divides line in ratio $m:n$m:n:$$\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}$$r=m+nmb+na​

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$\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}$r=m+nmb+na​

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📐 9. Dot Product (Scalar Product)

$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$$a⋅b=∣a∣∣b∣cosθ

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$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$a⋅b=∣a∣∣b∣cosθ

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📌 Component Form:

$$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$$a⋅b=a1​b1​+a2​b2​+a3​b3​

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📊 Properties:

• Commutative
• Distributive

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🔹 Angle Between Vectors

$$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$$cosθ=∣a∣∣b∣a⋅b​

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$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$cosθ=∣a∣∣b∣a⋅b​

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📌 Special Cases

• If $\vec{a} \cdot \vec{b} = 0$a⋅b=0 → perpendicular
• If $\vec{a} \cdot \vec{b} > 0$a⋅b>0 → acute angle

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✖️ 10. Cross Product (Vector Product)

$$\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \, \hat{n}$$a×b=∣a∣∣b∣sinθn^

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$\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \, \hat{n}$a×b=∣a∣∣b∣sinθn^

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📌 Determinant Form:

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$a×b=​i^a1​b1​​j^​a2​b2​​k^a3​b3​​​

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📊 Properties:

• Not commutative
• Distributive

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📐 Area of Parallelogram

$$|\vec{a} \times \vec{b}|$$∣a×b∣

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📐 Area of Triangle

$$\frac{1}{2} |\vec{a} \times \vec{b}|$$21​∣a×b∣

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📌 11. Scalar Triple Product

$$\vec{a} \cdot (\vec{b} \times \vec{c})$$a⋅(b×c)

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📊 Meaning:

• Volume of parallelepiped

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📌 Condition:

If scalar triple product = 0 → vectors are coplanar

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📌 12. Vector Triple Product

$$\vec{a} \times (\vec{b} \times \vec{c})$$a×(b×c)

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🔹 Formula:

$$= \vec{b}(\vec{a} \cdot \vec{c}) – \vec{c}(\vec{a} \cdot \vec{b})$$=b(a⋅c)−c(a⋅b)

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$\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}(\vec{a} \cdot \vec{c}) – \vec{c}(\vec{a} \cdot \vec{b})$a×(b×c)=b(a⋅c)−c(a⋅b)

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📊 13. Applications of Vectors

• Physics (force, velocity)
• Engineering
• Computer graphics
• Navigation
• 3D geometry

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❗ Common Mistakes

• Confusing dot and cross product
• Sign errors in determinants
• Forgetting unit vector
• Wrong angle calculation

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🧠 Exam Tips

• Practice vector formulas
• Learn identities
• Solve NCERT problems
• Focus on geometry applications

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📚 Practice Questions

1. Find magnitude of vector
2. Compute dot product
3. Find angle between vectors
4. Evaluate cross product

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🎯 Conclusion

Vector Algebra is a powerful mathematical tool that simplifies problems involving direction and magnitude. It is essential for understanding advanced topics in mathematics and physics.

Mastering vector operations ensures strong problem-solving skills in exams.