Define vector product of two vectors. Interpret it geometrically. Prove by vector method. Aa = Bb = Cc . [6]

Model Answer: Vector Product and Proof of the Sine Rule

1. Definition of Vector Product (Cross Product)

The vector product (or cross product) of two vectors $\vec{a}$ and $\vec{b}$ is a vector that is perpendicular to both $\vec{a}$ and $\vec{b}$. It is denoted by $\vec{a}\times \vec{b}$ and is defined as:

$$\vec{a}\times \vec{b}=|\vec{a}||\vec{b}|\sin \theta \hat{n}$$

where:

• $|\vec{a}|$ and $|\vec{b}|$ are the magnitudes of vectors $\vec{a}$ and $\vec{b}$.
• $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ (measured from $\vec{a}$ to $\vec{b}$).
• $\hat{n}$ is a unit vector perpendicular to the plane containing $\vec{a}$ and $\vec{b}$, following the right-hand rule.

Key Properties:

• Non-commutative: $\vec{a}\times \vec{b}=-(\vec{b}\times \vec{a})$.
• Magnitude: $|\vec{a}\times \vec{b}|=|\vec{a}||\vec{b}|\sin \theta$ (area of the parallelogram formed by $\vec{a}$ and $\vec{b}$).
• Direction: Given by the right-hand rule (thumb points in the direction of $\vec{a}\times \vec{b}$).

---

2. Geometric Interpretation

The vector product $\vec{a}\times \vec{b}$ represents:

1. A vector perpendicular to the plane containing $\vec{a}$ and $\vec{b}$.
2. Its magnitude is equal to the area of the parallelogram formed by $\vec{a}$ and $\vec{b}$.
3. Its direction follows the right-hand rule (if fingers curl from $\vec{a}$ to $\vec{b}$, the thumb points in the direction of $\vec{a}\times \vec{b}$).

Example: If $\vec{a}=\hat{i}$ and $\vec{b}=\hat{j}$, then: $$\vec{a}\times \vec{b}=\hat{i}\times \hat{j}=\hat{k}$$ Here, $\hat{k}$ is perpendicular to both $\hat{i}$ and $\hat{j}$, and its magnitude is 1 (since $\sin 90^{\circ }=1$).

---

3. Proof of the Sine Rule Using Vector Method

Statement to Prove: $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$

Given: A triangle ABC with sides a, b, c opposite to angles A, B, C respectively.

Proof:

Step 1: Position the Triangle in a Coordinate System

Place point A at the origin and side AB along the x-axis.

• Let $\overrightarrow{AB}=\vec{b}$ (vector from A to B).
• Let $\overrightarrow{AC}=\vec{c}$ (vector from A to C).

Step 2: Express Side $\overrightarrow{BC}$ as a Vector

$$\overrightarrow{BC}=\overrightarrow{AC}-\overrightarrow{AB}=\vec{c}-\vec{b}$$

Step 3: Compute the Magnitude of $\overrightarrow{BC}$ Using the Cross Product

The area of triangle ABC can be expressed in two ways:

1. Using the cross product: $$\text{Area}=\frac{1}{2}|\overrightarrow{AB}\times \overrightarrow{AC}|=\frac{1}{2}|\vec{b}\times \vec{c}|$$
2. Using the sine formula: $$\text{Area}=\frac{1}{2}bc\sin A$$

Equating the two expressions: $$|\vec{b}\times \vec{c}|=bc\sin A$$ But $|\vec{b}\times \vec{c}|=|\vec{b}||\vec{c}|\sin \theta$, where $\theta$ is the angle between $\vec{b}$ and $\vec{c}$. Since $\vec{b}$ and $\vec{c}$ are along sides AB and AC, the angle between them is A. Thus: $$|\vec{b}\times \vec{c}|=bc\sin A$$ This confirms consistency.

Step 4: Use the Law of Sines

Consider the vector $\overrightarrow{BC}=\vec{c}-\vec{b}$. Its magnitude is: $$|\overrightarrow{BC}|=a$$ Now, take the cross product of $\overrightarrow{BC}$ with $\vec{b}$: $$\overrightarrow{BC}\times \vec{b}=(\vec{c}-\vec{b})\times \vec{b}=\vec{c}\times \vec{b}-\vec{b}\times \vec{b}=\vec{c}\times \vec{b}$$ The magnitude is: $$|\overrightarrow{BC}\times \vec{b}|=|\vec{c}\times \vec{b}|=|\vec{c}||\vec{b}|\sin (\pi -B)=bc\sin B$$ But $|\overrightarrow{BC}\times \vec{b}|$ is also equal to: $$|\overrightarrow{BC}||\vec{b}|\sin \varphi$$ where $\varphi$ is the angle between $\overrightarrow{BC}$ and $\vec{b}$. From the geometry of the triangle, $\varphi =\pi -B$, so: $$ab\sin (\pi -B)=bc\sin B$$ Simplifying: $$a\sin B=b\sin A$$ $$\frac{\sin A}{a}=\frac{\sin B}{b}$$

Step 5: Repeat for Other Angles

Similarly, by taking the cross product of $\overrightarrow{BC}$ with $\vec{c}$, we get: $$\frac{\sin B}{b}=\frac{\sin C}{c}$$

Conclusion:

Combining the results: $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$

---

Final Answer:

The vector product (cross product) of two vectors $\vec{a}$ and $\vec{b}$ is a vector perpendicular to both, with magnitude $|\vec{a}||\vec{b}|\sin \theta$. Geometrically, it represents the area of the parallelogram formed by the two vectors. Using vector methods, we derived the sine rule for a triangle as: $$\boxed{\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}}$$