Understanding Vectors: Scalars, Representation, and Types

• Deals with vectors
• Overview of scalar and vector quantities, vector representation, vector types, angle directions, vector addition, and vector products

Scalar Quantities

• Fully determined by a number and its corresponding units.
• Examples include mass, time, length, volume, temperature, area, electric charge, and energy.

Vector Quantities

• Requires a number, units, direction, and sense to be fully determined.
• Examples include displacement, velocity, acceleration, force, weight, electric field, and momentum.

Graphical Representation of Vectors

• A vector is graphically represented by a directed segment (arrow).
• The segment's length represents the vector's magnitude (longer segment signifies greater magnitude).
• The line containing the segment determines the vector's direction.
• The arrowhead indicates the vector's sense.
• The vector's origin is also known as the point of application.

Types of Vectors

• Collinear vectors act along the same line of action.
• Concurrent vectors have lines of action that intersect at a point.
• Coplanar vectors lie in the same plane.
• Equal vectors possess the same magnitude, direction, and sense.
• Opposite vector has the same magnitude and direction as a given vector but with the opposite sense.
• Unit vector has a magnitude of 1.

Directional Angles

• Directional angles are the angles formed by the vector with the positive coordinate axes.
• Given a vector $\overrightarrow{A}$ in three-dimensional space, the directional angles $\alpha$, $\beta$, and $\gamma$ are the angles formed by the vector $\overrightarrow{A}$ with the x, y, and z axes, respectively.

Vector Addition

Graphical Method

• Polygon Method: Vectors are placed one after the other, maintaining their magnitude, direction, and sense. The resultant vector connects the origin of the first vector to the end of the last vector.
• Parallelogram Method: Vectors are placed with the same origin. Parallel lines are drawn to the vectors, forming a parallelogram. The resultant vector is the diagonal of the parallelogram that starts from the common origin of the vectors.

Analytical Method

• Sum of Collinear Vectors: The resultant equals the algebraic sum of the magnitudes, considering the sense of each vector.

• Sum of Concurrent Vectors: Vectors are decomposed into their rectangular components, which are then summed for each axis. The resultant vector has components equal to the sums obtained.

• A vector $\overrightarrow{A}$ in the Cartesian plane can be expressed as the sum of its rectangular components: $\overrightarrow{A} = A_x \hat{i} + A_y \hat{j}$, where $A_x$ and $A_y$ are the components of the vector $\overrightarrow{A}$ on the x and y axes, respectively, and $\hat{i}$ and $\hat{j}$ are the unit vectors in the directions of the x and y axes, respectively.

• The magnitude of the vector $\overrightarrow{A}$: $|\overrightarrow{A}| = \sqrt{A_x^2 + A_y^2}$

• The direction of the vector $\overrightarrow{A}$: $θ = tan^{-1}(\frac{A_y}{A_x})$, considering the signs of $A_x$ and $A_y$ to determine the correct quadrant.

• The sum of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$: $\overrightarrow{A} + \overrightarrow{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$

• The magnitude of the resulting vector: $|\overrightarrow{A} + \overrightarrow{B}| = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2}$

• The direction of the resulting vector: $θ = tan^{-1}(\frac{A_y + B_y}{A_x + B_x})$, considering the signs of $(A_x + B_x)$ and $(A_y + B_y)$ to determine the correct quadrant.

Vector Product

Scalar Product (Dot Product)

• The scalar product of two vectors is a scalar obtained by multiplying the magnitudes of the vectors by the cosine of the angle between them: $\overrightarrow{A} \cdot \overrightarrow{B} = |\overrightarrow{A}| |\overrightarrow{B}| cos(θ)$.
• Properties:
  • Commutative: $\overrightarrow{A} \cdot \overrightarrow{B} = \overrightarrow{B} \cdot \overrightarrow{A}$
  • Distributive: $\overrightarrow{A} \cdot (\overrightarrow{B} + \overrightarrow{C}) = \overrightarrow{A} \cdot \overrightarrow{B} + \overrightarrow{A} \cdot \overrightarrow{C}$
  • If $\overrightarrow{A} \cdot \overrightarrow{B} = 0$, the vectors are perpendicular.
• In terms of rectangular components: $\overrightarrow{A} \cdot \overrightarrow{B} = A_x B_x + A_y B_y + A_z B_z$

Vector Product (Cross Product)

• The vector product of two vectors is a vector perpendicular to the plane formed by the vectors. Its magnitude equals the product of the magnitudes of the vectors and the sine of the angle between them. Its direction is determined by the right-hand rule: $|\overrightarrow{A} \times \overrightarrow{B}| = |\overrightarrow{A}| |\overrightarrow{B}| sen(θ)$.
• The resulting vector's direction is perpendicular to the plane formed by vectors $\overrightarrow{A}$ and $\overrightarrow{B}$.
• The sense of the resulting vector is determined by the right-hand rule.
• Properties:
  • Non-commutative: $\overrightarrow{A} \times \overrightarrow{B} = -\overrightarrow{B} \times \overrightarrow{A}$
  • Distributive: $\overrightarrow{A} \times (\overrightarrow{B} + \overrightarrow{C}) = \overrightarrow{A} \times \overrightarrow{B} + \overrightarrow{A} \times \overrightarrow{C}$
  • If $\overrightarrow{A} \times \overrightarrow{B} = 0$, the vectors are parallel.
• In terms of rectangular components: $\overrightarrow{A} \times \overrightarrow{B} = (A_y B_z - A_z B_y) \hat{i} + (A_z B_x - A_x B_z) \hat{j} + (A_x B_y - A_y B_x) \hat{k}$
• Calculated using the determinant: $\overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix}$