Vectors

Understanding Vectors

  • A vector is a quantity that possesses both magnitude (size) and direction.

  • The length of the vector provides the magnitude whilst the direction is symbolised by the way the vector arrow is pointing.

  • Vectors are typically represented by lower case letters in bold or with an arrow above it, for example, a or →a.

  • Scalar quantities, in contrast to vectors, only have magnitude and no direction, for example, distance or speed.

Vector Algebra

  • Vector Addition: Two vectors can be added together to form a resultant vector. This can be done graphically using the ‘triangle rule’ or ‘parallelogram rule’, or algebraically where the individual components are added separately.

  • The ‘Triangle Rule’ for addition involves placing the vectors ‘head-to-tail’ and the resultant vector is the vector from the start of the first to the end of the second.

  • The ‘Parallelogram Rule’ for vector addition involves constructing a parallelogram using the two vectors as sides, the resultant vector is the diagonal of the parallelogram.

  • Vector Subtraction: This can be achieved by multiplying the vector being subtracted by -1 and then following the rules for vector addition.

Scalars and Vectors

  • A scalar quantity can be multiplied with a vector. This changes the magnitude of the vector but not its direction unless the scalar is negative, in which case the vector is also reversed.

  • The scalar product or dot product of two vectors gives a scalar quantity. It is found by multiplying the magnitudes of the vectors and the cosine of the angle between them.

  • The vector or cross product of two vectors gives a vector. The magnitude of this vector is found by multiplying the magnitudes of the products and the sine of the angle between them. The direction of the resulting vector is perpendicular to the plane containing the original vectors.

Unit Vectors

  • Unit vectors are vectors of length 1 and are used to define a direction.

  • Standard unit vectors in 3D are i, j and k where i represents one unit in the x-direction, j is one unit in the y direction and k is one unit in the z direction.

Applying Vectors

  • Vectors provide key insights for problems involving motion and force, including areas like engineering, physics and computer graphics.

  • For example, vector addition can be used to calculate the resultant force acting on a body, and understanding the properties of vectors is key to solving problems in statics and dynamics.

  • Practical applications include resultant forces, displacement, relative velocity - the list goes on, so a firm grounding in vectors is a crucial skill.