Multiplication of Vectors - Vector Product (Cross Product)

The Vector Product (or Cross Product) is a method of multiplying two vectors (\[\vec P\] and \[\vec Q\]) that results in a new vector (\[\vec R\]). This new vector is fundamentally related to the rotation or perpendicular effects created by the two original vectors.

The magnitude of the resulting vector R is defined by the product of the magnitudes of the two vectors and the sine of the smaller angle (θ) between them.
Magnitude: ∣R∣ = ∣ P × Q ∣ = PQ sin θ

Special Cases (Angle θ)

• Write both vectors with their i, j, k components (e.g. \[\mathbf{P}=P_x\hat{i}+P_y\hat{j}+P_z\hat{k}\]).
• Set up the cross product as a determinant using the components.
• Multiply as per the determinant formula:
  \[\mathbf{P}\times\mathbf{Q}=(P_yQ_z-P_zQ_y)\hat{i}+(P_zQ_x-P_xQ_z)\hat{j}+(P_xQ_y-P_yQ_x)\hat{k}\]
• Apply the right-hand rule to find the direction.
• Check for special cases like parallelism (result zero), perpendicular vectors (maximum result), or swapping order (direction reverses).

Fig. 2.10 (a): Vector product \[\vec R\] = \[\vec P\] × \[\vec Q\].

Fig. 2.10 (b): Vector product \[\vec S\] = \[\vec Q\] × \[\vec P\] .

• Used to calculate rotation and perpendicular effects in physics.
• Important for finding the torque and the magnetic force direction.
• Useful in 3D geometry, engineering, and vector analysis.
• Helps describe the angular velocity or velocity of rotating bodies.
• The cross product’s properties help distinguish physical laws (like non-commutativity).

\[\mathbf{r}=4\hat{i}+6\hat{j}-3\hat{k},\quad\mathbf{p}=2\hat{i}+4\hat{j}-5\hat{k}.\]
Compute \[\vec L\] = \[\vec r\] × \[\vec p\].

L = \[\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
4 & 6 & -3 \\
2 & 4 & -5
\end{vmatrix}\]

Simplify:

• i-component: (r_y × p_z) − (r_z × p_y) = (6 × (−5)) − ((−3) × 4) = -18\[\hat i\]
• j-component: (r_z × p_x) − (r_x × p_z) = ((−3) × 2) − (4 × (−5)) = +14\[\hat j\]
• k-component: (r_x × p_y) − (r_y × p_x) = (4 × 4) − (6 × 2) = +4\[\hat k\]

\[\vec L\] = ​-18\[\hat i\] + 14\[\hat j\] + 4 \[\hat k\]

Given A = \[5\hat{i}+6\hat{j}+4\hat{k}\], B = \[2\hat{i}-2\hat{j}+3\hat{k}\]. Find the angle between A and B:

1. Compute dot product: A⋅B = 10.
2. Compute magnitudes: ∣A∣ = \[\sqrt {77}\], ∣B∣ = \[\sqrt {17}\].
3. Calculate cos⁡ θ = \[\frac{10}{\sqrt{77}\sqrt{17}}\] ≈ 0.2764.
4. Find θ: θ = cos⁡⁻¹(0.2764) ≈ 73^∘58′.

Given \[\vec P\] = \[4\hat{i}-\hat{j}+8\hat{k}\], \[\vec Q\] = \[2\hat{i}-m\hat{j}+4\hat{k}.\]

Find: The value of m such that \[\vec P\] and \[\vec Q\] have the same direction

• For vectors to have the same direction, their components are proportional.
  \[\frac{P_x}{Q_x}=\frac{P_y}{Q_y}=\frac{P_z}{Q_z}.\]
• So \[\frac {4}{2}\] = \[\frac {-1}{-m}\] = \[\frac {8}{4}\] = 2.
  From \[\frac {-1}{-m}\] = 2 gives \[\frac {1}{m}\] = 2 ⇒ m = \[\frac {1}{2}\].
  Thus m = \[\frac {1}{2}\].

The cross product is essential for describing motion and forces involving rotation:

1. Torque: Bolts turn using a force and a handle, calculated as the cross product.
2. Magnetic Force: Moving charges in magnetic fields experience a force direction described by the cross product.
3. Rotating Fan Blade: The speed and direction of points on the blade use the cross product for velocity.