Vectors in 2D and 3D

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your ECAT exam.

A vector has both magnitude (length) and direction. A scalar has only magnitude. Vectors are essential in physics and engineering — force, velocity, acceleration, and electric fields are all vectors.

Vector Representation:

• In 2D: $\vec{A} = a\hat{i} + b\hat{j}$, magnitude $|\vec{A}| = \sqrt{a^2 + b^2}$
• In 3D: $\vec{A} = a\hat{i} + b\hat{j} + c\hat{k}$, magnitude $|\vec{A}| = \sqrt{a^2 + b^2 + c^2}$
• Unit vector in direction of $\vec{A}$: $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$

Basic Operations:

• Addition: $\vec{A} + \vec{B} = (a_1+a_2)\hat{i} + (b_1+b_2)\hat{j}$
• Subtraction: $\vec{A} - \vec{B} = (a_1-a_2)\hat{i} + (b_1-b_2)\hat{j}$
• Scalar multiplication: $k\vec{A} = ka\hat{i} + kb\hat{j}$ (scales magnitude by $|k|$, reverses if $k < 0$)

Dot (Scalar) Product: $$\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta = a_1a_2 + b_1b_2 + c_1c_2$$

• $\vec{A} \cdot \vec{B} = 0$ → perpendicular (orthogonal)
• $\vec{A} \cdot \vec{A} = |\vec{A}|^2$

Cross (Vector) Product: $$|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta$$ Direction by right-hand rule. In Cartesian: $$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \end{vmatrix}$$

⚡ ECAT exam tips:

• The cross product is NOT commutative: $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$
• The dot product IS commutative: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$
• $\vec{A} \cdot \vec{B} = 0$ means perpendicular (cos 90° = 0)
• $|\vec{A} \times \vec{B}| = 0$ means parallel (sin 0° = 0)

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🟡 Standard — Regular Study (2d–2mo)

For ECAT students who want genuine understanding of vectors.

Scalar Triple Product:

$$\vec{A} \cdot (\vec{B} \times \vec{C}) = \begin{vmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{vmatrix}$$

Properties:

• Cyclic permutation doesn’t change value: $\vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{B} \cdot (\vec{C} \times \vec{A}) = \vec{C} \cdot (\vec{A} \times \vec{B})$
• If $\vec{A} \cdot (\vec{B} \times \vec{C}) = 0$, the three vectors are coplanar

Vector Triple Product: $$\vec{A} \times (\vec{B} \times \vec{C}) = \vec{B}(\vec{A} \cdot \vec{C}) - \vec{C}(\vec{A} \cdot \vec{B})$$

This is an important identity. Note that it’s NOT associative: $\vec{A} \times (\vec{B} \times \vec{C}) \neq (\vec{A} \times \vec{B}) \times \vec{C}$.

Angle Between Two Vectors: $$\cos\theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|} = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$$

Projection of One Vector onto Another:

The scalar projection (component) of $\vec{A}$ onto $\vec{B}$: $$\text{comp}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|}$$

The vector projection of $\vec{A}$ onto $\vec{B}$: $$\text{proj}_{\vec{B}} \vec{A} = \left(\frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2}\right) \vec{B}$$

⚡ Common student mistakes:

1. Confusing dot product and cross product — dot product gives a scalar, cross product gives a vector
2. Forgetting the right-hand rule for cross product direction
3. Using the wrong sign when calculating cross product determinants
4. For scalar triple product: not recognizing that it gives the volume of the parallelepiped formed by three vectors

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for ECAT mastery of vector algebra.

Vector Addition — Triangle and Parallelogram Laws:

Triangle law: To add $\vec{A}$ and $\vec{B}$, place the tail of $\vec{B}$ at the head of $\vec{A}$. The resultant $\vec{R} = \vec{A} + \vec{B}$ is the vector from the tail of $\vec{A}$ to the head of $\vec{B}$.

Parallelogram law: Place $\vec{A}$ and $\vec{B}$ with common tail. Complete the parallelogram. The resultant is the diagonal from the common tail.

Magnitude of sum and difference: $$|\vec{A} + \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2\vec{A}\cdot\vec{B}$$ $$|\vec{A} - \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 - 2\vec{A}\cdot\vec{B}$$

Direction Cosines:

For a vector $\vec{A} = a\hat{i} + b\hat{j} + c\hat{k}$: Direction cosines: $l = \cos\alpha = \frac{a}{|\vec{A}|}$, $m = \cos\beta = \frac{b}{|\vec{A}|}$, $n = \cos\gamma = \frac{c}{|\vec{A}|}$ Note: $l^2 + m^2 + n^2 = 1$.

Line and Plane in Vector Form:

Equation of a line passing through point with position vector $\vec{a}$ with direction $\vec{b}$: $$\vec{r} = \vec{a} + t\vec{b}$$ or parametric: $x = a_1 + tb_1$, $y = a_2 + tb_2$, $z = a_3 + tb_3$.

Equation of a plane passing through point $\vec{a}$ with normal $\vec{n}$: $$\vec{n} \cdot (\vec{r} - \vec{a}) = 0$$

Distance from a point to a line (3D): For point $P$ to line through $A$ with direction $\vec{b}$: $$d = \frac{|\vec{b} \times (\vec{A} - \vec{P})|}{|\vec{b}|}$$

Distance from a point to a plane: For point $(x_1,y_1,z_1)$ to plane $ax + by + cz + d = 0$: $$d = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$$

Coplanarity of Four Points:

Four points $A, B, C, D$ (with position vectors $\vec{a}, \vec{b}, \vec{c}, \vec{d}$) are coplanar if: $$(\vec{b} - \vec{a}) \cdot [(\vec{c} - \vec{a}) \times (\vec{d} - \vec{a})] = 0$$

Work Done by a Force:

Work done by force $\vec{F}$ moving an object from position $\vec{r_1}$ to $\vec{r_2}$: $$W = \vec{F} \cdot (\vec{r_2} - \vec{r_1})$$

Angular Velocity:

A particle rotating with angular velocity $\vec{\omega}$ has linear velocity: $$\vec{v} = \vec{\omega} \times \vec{r}$$ This is a key result in rotational mechanics.

ECAT Previous Year Patterns:

• Dot product and cross product: very common
• Angle between vectors: common
• Magnitude calculations: common
• Scalar and vector projections: periodic
• 3D vector geometry: occasionally tested

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