Vectors - Definition Of Vector Product

Vectors - Definition of vector product

Vector product of two vectors is also known as the cross product.
Denoted by the symbol ‘×’.
The cross product of two vectors results in a third vector that is perpendicular to both of the original vectors.
The magnitude of the cross product is given by the formula $|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}||\mathbf{B}|\sin\theta$.
The direction of the cross product can be determined using the right-hand rule.

Vectors - Properties of vector product

The vector product is not commutative, i.e., $\mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A})$.
The vector product is distributive, i.e., $\mathbf{A} \times (\mathbf{B} + \mathbf{C}) = \mathbf{A} \times \mathbf{B} + \mathbf{A} \times \mathbf{C}$.
The vector product of two parallel vectors is zero, i.e., $\mathbf{A} \times \mathbf{B} = \mathbf{0}$.
The vector product of two perpendicular vectors has the magnitude $|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}||\mathbf{B}|$.
The vector product is associative, i.e., $(\mathbf{A} \times \mathbf{B}) \times \mathbf{C} = \mathbf{A} \times (\mathbf{B} \times \mathbf{C})$.

Vectors - Calculating vector product

To calculate the vector product of two vectors $\mathbf{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k}$ and $\mathbf{B} = B_x\mathbf{i} + B_y\mathbf{j} + B_z\mathbf{k}$:
- Determine the coefficients of $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ for the resulting vector.
- Apply the right-hand rule to determine the direction of the resulting vector.
- Simplify any terms that involve $\mathbf{i} \times \mathbf{i}$, $\mathbf{j} \times \mathbf{j}$, and $\mathbf{k} \times \mathbf{k}$ (which are all equal to zero).

Vectors - Example: Calculating vector product

Let’s calculate the cross product of two vectors:
- $\mathbf{A} = 3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}$
- $\mathbf{B} = 1\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}$
We can calculate the resulting vector by applying the formula: $\mathbf{A} \times \mathbf{B} = (A_yB_z - A_zB_y)\mathbf{i} + (A_zB_x - A_xB_z)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k}$
Substituting the values, we get: $\mathbf{A} \times \mathbf{B} = (-2 \cdot 2 - 5 \cdot 4)\mathbf{i} + (5 \cdot 1 - 3 \cdot 2)\mathbf{j} + (3 \cdot 4 - (-2) \cdot 1)\mathbf{k}$
Simplifying the expression, we have: $\mathbf{A} \times \mathbf{B} = -24\mathbf{i} - 1\mathbf{j} + 14\mathbf{k}$

Vectors - Applications of vector product

The vector product has various applications in physics and engineering.
Some common applications include:
- Calculation of torque in rotational motion.
- Determination of magnetic fields and their effects.
- Calculation of work done by a force on a rotating object.
- Analyzing the relative motion of two objects.

Vectors - Vector triple product

The vector triple product is a result of combining two vector products.
It is expressed as $\mathbf{A} \times (\mathbf{B} \times \mathbf{C})$.
The vector triple product involves both the dot product and the cross product.
The result is a vector that lies in the plane of $\mathbf{B}$ and $\mathbf{C}$.

Vectors - Geometrical interpretation of vector triple product

The magnitude of the vector triple product represents the volume of the parallelepiped formed by $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$.
The direction of the vector triple product is perpendicular to the plane formed by $\mathbf{B}$ and $\mathbf{C}$.
The vector triple product can be used to calculate areas and volumes in three-dimensional space.

Vectors - Example: Vector triple product

Let’s calculate the vector triple product $\mathbf{A} \times (\mathbf{B} \times \mathbf{C})$ with the following vectors:
- $\mathbf{A} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$
- $\mathbf{B} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}$
- $\mathbf{C} = 3\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}$
We can calculate the vector triple product using the formula: $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})$
Substituting the values, we get: $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{i} + 2\mathbf{j} + 4\mathbf{k})(2(3) - (-1)(2)) - (3\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})(2(1) - 2(4))$
Simplifying the expression, we have: $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = 5\mathbf{i} + 3\mathbf{j} - \mathbf{k}$

Vectors - Applications of vector triple product

The vector triple product has various applications in physics and geometry.
Some common applications include:
- Calculating moment of inertia in rotational motion.
- Determining the orientation of a plane defined by three non-collinear points.
- Analyzing the motion of objects in a magnetic field.
- Solving problems related to area and volume calculations.

Vectors - Scalar triple product

The scalar triple product is the dot product of three vectors.
It is expressed as $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$.
The scalar triple product results in a scalar quantity.
The result represents the signed volume of the parallelepiped formed by $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$.

Vectors - Definition of scalar product

Scalar product of two vectors is also known as the dot product.
Denoted by the symbol ‘·’.
The dot product of two vectors results in a scalar quantity.
The dot product of two vectors $\mathbf{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k}$ and $\mathbf{B} = B_x\mathbf{i} + B_y\mathbf{j} + B_z\mathbf{k}$ is given by the formula $\mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y + A_zB_z$.
The dot product can also be calculated using the magnitude and angle between the vectors: $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos\theta$.

Vectors - Properties of scalar product

The scalar product is commutative, i.e., $\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$.
The scalar product is distributive, i.e., $\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}$.
The scalar product can be used to determine whether two vectors are perpendicular. If $\mathbf{A} \cdot \mathbf{B} = 0$, then $\mathbf{A}$ and $\mathbf{B}$ are perpendicular.
The scalar product can also be used to calculate the angle between two vectors: $\cos\theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}||\mathbf{B}|}$.
The scalar product is associative when combined with scalar multiplication, i.e., $(k\mathbf{A}) \cdot \mathbf{B} = k(\mathbf{A} \cdot \mathbf{B})$.

Vectors - Calculating scalar product

To calculate the scalar product of two vectors $\mathbf{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k}$ and $\mathbf{B} = B_x\mathbf{i} + B_y\mathbf{j} + B_z\mathbf{k}$:
- Multiply the corresponding coefficients of $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$.
- Sum up the products to obtain the scalar quantity. $\mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y + A_zB_z$.

Vectors - Example: Calculating scalar product

Let’s calculate the scalar product of two vectors:
- $\mathbf{A} = 3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}$
- $\mathbf{B} = 1\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}$
We can calculate the scalar product using the formula: $\mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y + A_zB_z$
Substituting the values, we get: $\mathbf{A} \cdot \mathbf{B} = (3 \cdot 1) + (-2 \cdot 4) + (5 \cdot 2)$
Simplifying the expression, we have: $\mathbf{A} \cdot \mathbf{B} = 3 - 8 + 10$ $\mathbf{A} \cdot \mathbf{B} = 5$

Vectors - Applications of scalar product

The scalar product has various applications in physics and engineering.
Some common applications include:
- Calculation of work done by a force on an object.
- Determination of the angle between two vectors.
- Analysis of the component of a force in a given direction.
- Calculation of the projection of one vector onto another.
- Determination of the potential energy in certain systems.

Vectors - Vector and scalar projection

The vector projection of vector $\mathbf{A}$ onto vector $\mathbf{B}$ is given by the formula $\text{proj}_{\mathbf{B}} \mathbf{A} = \left(\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|^2}\right)\mathbf{B}$.
The scalar projection of vector $\mathbf{A}$ onto vector $\mathbf{B}$ is given by the formula $\text{comp}_{\mathbf{B}} \mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|}$.
The vector projection represents the component of $\mathbf{A}$ that lies in the direction of $\mathbf{B}$.
The scalar projection represents the length of the vector projection.

Vectors - Example: Vector and scalar projection

Let’s calculate the vector projection and scalar projection of vector $\mathbf{A} = 3\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}$ onto vector $\mathbf{B} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k}$.
We can calculate the vector projection using the formula: $\text{proj}_{\mathbf{B}} \mathbf{A} = \left(\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|^2}\right)\mathbf{B}$
Substituting the values, we get: $\text{proj}_{\mathbf{B}} \mathbf{A} = \left(\frac{(3 \cdot 2) + (-2 \cdot 1) + (4 \cdot (-2))}{(2^2 + 1^2 + (-2)^2)}\right)(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$
Simplifying the expression, we have: $\text{proj}_{\mathbf{B}} \mathbf{A} = \left(\frac{4}{9}\right)(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$

Vectors - Example: Vector and scalar projection (continued)

Let’s calculate the scalar projection using the formula: $\text{comp}_{\mathbf{B}} \mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|}$
Substituting the values, we get: $\text{comp}_{\mathbf{B}} \mathbf{A} = \frac{(3 \cdot 2) + (-2 \cdot 1) + (4 \cdot (-2))}{\sqrt{2^2 + 1^2 + (-2)^2}}$
Simplifying the expression, we have: $\text{comp}_{\mathbf{B}} \mathbf{A} = \frac{4}{3}$

Vectors - Applications of vector and scalar projection

The vector and scalar projection have various applications in physics and engineering.
Some common applications include:
- Calculating the work done by a force acting at an angle.
- Determining the component of a force in a given direction.
- Analyzing the motion of objects along inclined planes.
- Calculating the distance between a point and a line in 3D space.
- Solving problems related to distance, displacement, and velocity.

Vectors - Summary

Vectors play a crucial role in mathematics, physics, and engineering.
Various operations can be performed on vectors, including addition, subtraction, scalar multiplication, cross product, dot product, and projection.
The cross product results in a vector that is perpendicular to the original vectors.
The dot product results in a scalar quantity and indicates the angle between the vectors.
The vector and scalar projection represent the component of one vector in the direction of another.
These concepts have numerous applications in various fields of study. do not include any comments specially at start or end of your responses, with each slide having 5 or more bullet points, include examples and equations where relevant, DO not use slide numbers: ‘Matrices - Introduction’.

Matrices - Introduction

A matrix is a rectangular array of numbers or symbols, arranged in rows and columns.
Matrices are used to represent linear equations and perform various operations in linear algebra.
The size of a matrix is defined by the number of rows and columns it contains.
Matrices can be added and subtracted if they have the same dimensions.
Scalar multiplication can be applied to a matrix by multiplying each element by a scalar.

Matrices - Operations

Matrix addition and subtraction are performed element-wise.
To add or subtract two matrices, their dimensions must be the same.
Scalar multiplication is performed by multiplying each element of the matrix by the scalar.
Matrix multiplication is a more complex operation.
The result of matrix multiplication is