Vectors Short Notes for Engineering Entrance Exams | Definitions & Formulas

VECTORS — SHORT NOTES

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1. Basics of Vectors

Scalar: Quantity with only magnitude

Examples → mass, time, speed, temperature.

Vector: Quantity with magnitude and direction

Examples → displacement, velocity, acceleration, force, momentum.

Representation:

$$\vec{A}=A_x\widehat{i}+A_y\widehat{j}+A_z\widehat{k}$$

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2. Types of Vectors

• Unit vector:

  $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$

• Null/Zero vector → magnitude 0

• Equal vectors → same magnitude & direction

• Collinear / Parallel vectors

• Negative vectors → opposite directions

• Coplanar vectors → lie in same plane

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3. Magnitude of a Vector

$$\mathrm{\mid }\vec{A}\mathrm{\mid }=\sqrt{A_x^2+A_y^2+A_z^2}$$

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4. Addition & Subtraction

Parallelogram law:

$$\vec{R} = \vec{A} + \vec{B}$$
$$R=\sqrt{A^2+B^2+2AB\cos \theta }$$

Triangle law:

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

Subtraction:

$$\vec{A}-\vec{B}=\vec{A}+(-\vec{B})$$

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5. Scalar (Dot) Product

$$\vec{A} \cdot \vec{B} = AB\cos\theta$$

Useful results:

• If A ⊥ B → dot product = 0

• Work done = $\vec{F} \cdot \vec{s}$

• In component form:

$$\vec{A}\cdot \vec{B}=A_x{B}_x+A_y{B}_y+A_z{B}_{\mathrm{z}}$$

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6. Vector (Cross) Product

$$\vec{A}\times \vec{B}=AB\sin \theta \widehat{n}$$

(where (\hat{n}) = unit vector perpendicular to plane)

Important:

• If A ∥ B → cross product = 0

• Torque,

• T= $\vec{r} \times \vec{F}$
  

• Area of parallelogram = |A × B|

• Area of triangle = ½ |A × B|

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7. Scalar Triple Product (STP)

$$\vec{A}\cdot (\vec{B}\times \vec{C})$$

• Represents volume of a parallelepiped.

• If STP = 0 → vectors are coplanar.

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8. Vector Triple Product

$$\vec{A}\times (\vec{B}\times \vec{C})=(\vec{A}\cdot \vec{C})\vec{B}-(\vec{A}\cdot \vec{B})\vec{C}$$

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9. Direction Cosines & Direction Ratios

Direction cosines (l, m, n):

$$l=\frac{A_x}{A},m=\frac{A_y}{A},n=\frac{A_z}{\mathrm{A<span\; class="vlist"><span\; class="mord"><span\; class="mord"><span\; class="msupsub"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist-s"></span></span><span\; class="vlist-r"><span\; class="vlist"></span></span></span></span></span></span></span><span\; class="vlist-s"></span>}}$$$$l^2 + m^2 + n^2 = 1$$

Direction ratios:

Numbers proportional to the components of the vector.

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10. Position Vector & Displacement Vector

• Position vector of point P(x, y, z):

$$\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$$

• Displacement from A(x₁, y₁, z₁) to B(x₂, y₂, z₂):

$$\vec{AB} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}$$

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11. Important Identities

• $\hat{i} \times \hat{j} = \hat{k}$
  

• $\hat{j} \times \hat{k} = \hat{i}$
  

• $\hat{k} \times \hat{i} = \hat{j}$

• $\widehat{i}\cdot \widehat{j}=\widehat{j}\cdot \widehat{k}=\widehat{k}\cdot \widehat{i}=0$

• $\widehat{i}\cdot \widehat{i}=\widehat{j}\cdot \widehat{j}=\widehat{k}\cdot \widehat{k}=1$

12. Common Exam Questions

1. Given vectors, find magnitude, unit vector.

2. Direction cosines, projection of a vector.

3. Dot/cross product numerical.

4. Find area using |A × B|.

5. Volume using triple product.

6. Condition for collinearity / coplanarity.