3.4 Q-4

Exercise 3.4 — Question 4

Topic: Cross Product (Vector Product) of Two Vectors

This question applies the cross product to find the angle between two vectors and to compute the vector product.

---

Key Formula: Cross Product

For two vectors $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$, the cross product is:

$\vec{a}\times \vec{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$

Expanding: $\vec{a}\times \vec{b}=(a_2{b}_3-a_3{b}_2)\hat{i}-(a_1{b}_3-a_3{b}_1)\hat{j}+(a_1{b}_2-a_2{b}_1)\hat{k}$

---

Magnitude of Cross Product

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

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

To find the angle using cross product: $\sin \theta =\frac{|\vec{a}\times \vec{b}|}{|\vec{a}||\vec{b}|}$

$\theta ={\sin }^{-1}\left(\frac{|\vec{a}\times \vec{b}|}{|\vec{a}||\vec{b}|}\right)$

---

Geometrical Interpretation

The cross product $\vec{a}\times \vec{b}$ is a vector that is:

• Perpendicular to both $\vec{a}$ and $\vec{b}$
• Has magnitude equal to the area of the parallelogram formed by $\vec{a}$ and $\vec{b}$: $\text{Area}=|\vec{a}\times \vec{b}|$

---

Properties of Cross Product

---

Worked Example

Find $\vec{a}\times \vec{b}$ and the angle between them, given: $\vec{a}=2\hat{i}+3\hat{j}-\hat{k},\vec{b}=\hat{i}-2\hat{j}+3\hat{k}$

Step 1: Compute the cross product $\vec{a}\times \vec{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& -1\\ 1& -2& 3\end{array}\right|$

$=\hat{i}(3\cdot 3-(-1)(-2))-\hat{j}(2\cdot 3-(-1)(1))+\hat{k}(2(-2)-3\cdot 1)$

$=\hat{i}(9-2)-\hat{j}(6+1)+\hat{k}(-4-3)$

$=7\hat{i}-7\hat{j}-7\hat{k}$

Step 2: Find magnitudes $|\vec{a}\times \vec{b}|=\sqrt{49+49+49}=7\sqrt{3}$

$|\vec{a}|=\sqrt{4+9+1}=\sqrt{14},|\vec{b}|=\sqrt{1+4+9}=\sqrt{14}$

Step 3: Find the angle $\sin \theta =\frac{7\sqrt{3}}{\sqrt{14}\cdot \sqrt{14}}=\frac{7\sqrt{3}}{14}=\frac{\sqrt{3}}{2}$

$\theta ={\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)=60°$

---