5.5. Operations on Vectors

1. Addition of Vectors:

   • Graphical Method: Vectors are added head-to-tail. If two vectors 𝐴⃗ and 𝐵⃗ are represented as arrows, the sum of the vectors is represented by the diagonal of the parallelogram formed by the two vectors.
   • Algebraic Method: In component form, the sum of two vectors is obtained by adding their corresponding components.

   $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}$

In vector addition, there are two main methods: graphical and algebraic.

1. Graphical Method: Imagine two arrows drawn on a paper - one for each vector (𝐴⃗ and 𝐵⃗). To add these vectors, place the tail of the second vector (𝐵⃗) at the head of the first vector (𝐴⃗). The resulting vector, which points from the tail of the first vector to the head of the second vector, represents their sum.

2. Algebraic Method: Each vector can be broken down into its components along the x, y, and z axes. By naming the components of vector 𝐴⃗ as (A_x, A_y, A_z) and those of vector 𝐵⃗ as (B_x, B_y, B_z), we simply add the corresponding components. This gives us a new vector which is a combination of the two.

Think of vector addition like finding the resultant path of two journeys. If you walk 3 meters north and then 4 meters east, your total displacement can be represented as a straight line from your starting point to the endpoint. You can visualize this as forming a right triangle where one leg is the distance walked north, and the other leg is the distance walked east. The hypotenuse represents the direct path, or the sum of your vector movements.

2. Subtraction of Vectors: The difference of two vectors is obtained by reversing the direction of the second vector and then adding them.

   $\vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j} + (A_z - B_z)\hat{k}$

Vector subtraction involves two main steps. First, you take the vector you want to subtract (𝐵⃗) and reverse its direction. Once the direction is reversed, you treat this reversed vector as if it were being added to the first vector (𝐴⃗).

Using components, if you have two vectors 𝐴⃗ and 𝐵⃗, their subtraction results in a new vector where each component from 𝐵⃗ is subtracted from the corresponding component in 𝐴⃗.

Imagine tracking your progress while jogging. If you started at point A, jogged to point B, and then decided to return to point A, the two legs of your journey represent vector movement. Subtracting the vector pointing from A to B means you need to take the reverse path from B back to A. Thus, subtraction in vector terms can be understood as reversing your movement.

3. Scalar Multiplication: A vector can be multiplied by a scalar (a real number), which affects the magnitude of the vector but not its direction (unless the scalar is negative).

   $k \cdot \vec{A} = k \cdot (A_x \hat{i} + A_y \hat{j} + A_z \hat{k})$

In scalar multiplication, a vector is multiplied by a real number (which is called a scalar). This operation will change the magnitude of the vector but will retain its direction unless the scalar is negative. If the scalar is negative, the direction of the resulting vector is reversed.

For example, multiplying vector 𝐴⃗ by 2 will make it twice as long in the same direction. Conversely, multiplying it by -1 will keep the same length but flip the direction.

Consider a car moving in a straight line. If your velocity is a vector representing speed in a direction (say, 60 km/h to the east), multiplying that vector by a scalar of ½ (0.5) would indicate moving at 30 km/h to the east, while multiplying by -1 indicates moving at 60 km/h but to the west.

4. Dot Product (Scalar Product): The dot product of two vectors 𝐴⃗ and 𝐵⃗ is a scalar quantity given by: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ Alternatively, it can be written as: $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta$ where $\theta$ is the angle between the two vectors.

The dot product of two vectors gives a single scalar value that reflects how much one vector extends in the direction of another. To compute the dot product algebraically, multiply the corresponding components of the two vectors and sum these products.

The dot product can also be expressed using the magnitude of the vectors and the cosine of the angle between them. This helps to determine the angle between the two vectors, where a dot product of zero means the vectors are perpendicular.

Think of the dot product as measuring how much 'shadow' one vector casts on another. Light shining at an angle to a surface illustrates this: where the light is almost parallel, the mess of shadows aligns closely, resulting in a large dot product. Conversely, light coming from straight above a flat object yields little to no shadow, demonstrating a smaller product.

5. Cross Product (Vector Product): The cross product of two vectors 𝐴⃗ and 𝐵⃗ results in a vector that is perpendicular to both 𝐴⃗ and 𝐵⃗, and is given by: $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ The magnitude of the cross product is given by: $|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta$ where $\theta$ is the angle between the two vectors.

The cross product results in a new vector that is orthogonal (perpendicular) to the plane formed by the two original vectors. This operation can be computed using the determinant of a matrix formed by the unit vectors and the components of the vectors A and B. Its magnitude depends on the angle between the two vectors, and demonstrates how much area is covered by the parallelogram formed by these two vectors.

Picture holding two pens at intersecting angles: the tension between the two pens creates a 'twisting' force that 'points' outwards in a new direction perpendicular to the flat surface you created between them. The cross product can be visualized as producing force or motion resulting from combining two circular movements in an object.

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