1.3 Vector algebra and its applications in mechanics

Vector algebra is the backbone of mechanics, allowing us to represent forces and moments mathematically. It's all about arrows with size and direction, helping us solve real-world physics problems. We'll learn to add, subtract, and break down these arrows to understand how objects behave under different forces.

This knowledge is crucial for analyzing structures and machines in engineering. We'll use vector operations to find resultant forces and moments, determine equilibrium conditions, and solve complex mechanical systems. It's the foundation for tackling more advanced topics in statics and dynamics.

Vectors and their properties

Vector definition and representation

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• Vectors are mathematical objects that possess both magnitude and direction, typically represented by arrows
• The magnitude of a vector is the length of the arrow, indicating the size or intensity of the quantity (force, velocity, displacement)
• The direction of a vector is specified by the orientation of the arrow, denoting the line of action along which the quantity acts
• Vectors are used to represent physical quantities that have both magnitude and direction (force, velocity, displacement)

Equality of vectors

• Two vectors are considered equal if and only if they have the same magnitude and direction, regardless of their starting points
• Vectors with the same magnitude and direction can be freely translated in space without changing their properties
• Equal vectors have identical effects on a system, regardless of their location

Vector operations

Vector addition and subtraction

• Vector addition combines two or more vectors to obtain a resultant vector, following the parallelogram law or the triangle rule
  • The parallelogram law states that the resultant vector is the diagonal of a parallelogram formed by the two vectors being added, with their tails coinciding
  • The triangle rule states that the resultant vector is obtained by placing the tail of the second vector at the head of the first vector, and drawing the resultant from the tail of the first to the head of the second
• Vector subtraction finds the difference between two vectors by adding the negative of the vector being subtracted to the other vector
  • To subtract vector B from vector A, add the negative of vector B to vector A
  • The negative of a vector has the same magnitude but opposite direction

Scalar multiplication and vector products

• Scalar multiplication of a vector involves multiplying the magnitude of the vector by a scalar value, while maintaining or reversing the direction
  • If the scalar is positive, the vector's direction remains the same
  • If the scalar is negative, the vector's direction is reversed
• The dot product (scalar product) of two vectors is a scalar value obtained by multiplying their magnitudes and the cosine of the angle between them
  • The dot product represents the projection of one vector onto the other
  • The dot product is commutative and distributive over vector addition
• The cross product (vector product) of two vectors is a vector perpendicular to the plane containing the two vectors
  • The magnitude of the cross product is equal to the product of the vectors' magnitudes and the sine of the angle between them
  • The direction of the cross product is determined by the right-hand rule
  • The cross product is anticommutative and distributive over vector addition

Resolving vectors into components

Vector resolution

• Vector resolution breaks a vector into its component parts, typically along the x, y, and z axes of a coordinate system
• The x-component of a vector is the projection of the vector onto the x-axis, calculated by multiplying the vector's magnitude by the cosine of the angle between the vector and the positive x-axis
• The y-component of a vector is the projection of the vector onto the y-axis, calculated by multiplying the vector's magnitude by the cosine of the angle between the vector and the positive y-axis
• The z-component of a vector is the projection of the vector onto the z-axis, calculated by multiplying the vector's magnitude by the cosine of the angle between the vector and the positive z-axis

Calculating magnitude and direction from components

• The magnitude of a vector can be calculated from its components using the Pythagorean theorem in 2D or 3D space
  • In 2D: $magnitude = \sqrt{x^2 + y^2}$
  • In 3D: $magnitude = \sqrt{x^2 + y^2 + z^2}$
• The direction of a vector can be determined from its components using the arctangent function (atan2) or by calculating the angles with respect to the coordinate axes
  • In 2D: $\theta = atan2(y, x)$, where $\theta$ is the angle between the vector and the positive x-axis
  • In 3D, the angles with respect to the coordinate axes can be found using the inverse cosine function (arccos)

Vector algebra in physics problems

Forces as vectors

• Forces can be represented as vectors, with the magnitude representing the intensity of the force and the direction indicating the line of action
• The resultant force acting on an object can be determined by adding all the force vectors acting on the object using vector addition
• Equilibrium of an object occurs when the resultant force acting on the object is zero
  • In 2D: $\sum F_x = 0$ and $\sum F_y = 0$
  • In 3D: $\sum F_x = 0$, $\sum F_y = 0$, and $\sum F_z = 0$

Moments as vectors

• Moments (or torques) can be represented as vectors, with the magnitude equal to the product of the force and the perpendicular distance from the line of action to the point of rotation
• The resultant moment acting on an object can be determined by adding all the moment vectors acting on the object using vector addition
• The concept of vector cross products is used to calculate moments, as the moment vector is the cross product of the position vector (from the point of rotation to the point of force application) and the force vector
  • $\vec{M} = \vec{r} \times \vec{F}$, where $\vec{M}$ is the moment vector, $\vec{r}$ is the position vector, and $\vec{F}$ is the force vector
• Equilibrium of an object occurs when the resultant moment acting on the object is zero
  • In 2D: $\sum M = 0$
  • In 3D: $\sum M_x = 0$, $\sum M_y = 0$, and $\sum M_z = 0$