Products of Vectors

Products of Vectors

Products of Vectors

Introduction

Vector products allow us to combine vectors in different ways, depending on whether we need a scalar result (dot product) or a vector result (cross product). These operations are essential in physics and engineering for understanding quantities like work, torque, and angular momentum.

Dot Product (Scalar Product)

The dot product of two vectors \vec{A} and \vec{B} gives a scalar value. It is defined as:

    \[\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta\]

Here:

  • |\vec{A}|: Magnitude of \vec{A}.
  • |\vec{B}|: Magnitude of \vec{B}.
  • \theta: Angle between \vec{A} and \vec{B}.

If \vec{A} and \vec{B} are expressed in components, the dot product is:

    \[\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z\]

Example: If \vec{A} = 2 \hat{i} + 3 \hat{j} and \vec{B} = \hat{i} + 4 \hat{j}:

    \[\vec{A} \cdot \vec{B} = (2)(1) + (3)(4) = 2 + 12 = 14\]

The dot product is 14.

Cross Product (Vector Product)

The cross product of two vectors \vec{A} and \vec{B} gives a vector that is perpendicular to both \vec{A} and \vec{B}. It is defined as:

    \[\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin\theta \, \hat{n}\]

Here:

  • |\vec{A}| and |\vec{B}|: Magnitudes of \vec{A} and \vec{B}.
  • \theta: Angle between \vec{A} and \vec{B}.
  • \hat{n}: Unit vector perpendicular to both \vec{A} and \vec{B}, determined by the right-hand rule.

If \vec{A} and \vec{B} are expressed in components, the cross product is calculated using the determinant:

    \[\vec{A} \times \vec{B} = \begin{vmatrix}         \hat{i} & \hat{j} & \hat{k} \\         A_x & A_y & A_z \\         B_x & B_y & B_z \\         \end{vmatrix}\]

Example: If \vec{A} = 2 \hat{i} + 3 \hat{j} + \hat{k} and \vec{B} = \hat{i} + 4 \hat{j} + 2 \hat{k}:

    \[\vec{A} \times \vec{B} = \begin{vmatrix}         \hat{i} & \hat{j} & \hat{k} \\         2 & 3 & 1 \\         1 & 4 & 2 \\         \end{vmatrix}\]

Expanding the determinant:

    \[\vec{A} \times \vec{B} = \hat{i}(3 \cdot 2 - 1 \cdot 4) - \hat{j}(2 \cdot 2 - 1 \cdot 1) + \hat{k}(2 \cdot 4 - 3 \cdot 1)\]

    \[\vec{A} \times \vec{B} = \hat{i}(6 - 4) - \hat{j}(4 - 1) + \hat{k}(8 - 3)\]

    \[\vec{A} \times \vec{B} = 2 \hat{i} - 3 \hat{j} + 5 \hat{k}\]

The cross product is \vec{A} \times \vec{B} = 2 \hat{i} - 3 \hat{j} + 5 \hat{k}.

Applications of Vector Products

  • Dot Product: Used to calculate work done (W = \vec{F} \cdot \vec{d}) and projection of vectors.
  • Cross Product: Used to calculate torque (\vec{\tau} = \vec{r} \times \vec{F}) and angular momentum.

Key Takeaways

Dot products result in scalars, while cross products result in vectors. Both operations are crucial for analyzing physical phenomena, such as forces, motion, and energy.

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