Learning Outcomes: and may be used to represent (a) State the types of vectors. (b) Find the magnitude of a vector and unit vector. (c) Perform addition, subtraction and scalar multiplication of vectors. At eth end of the lesson, students should be able to Scalars & Vectors Scalars Vectors •Also known as scalar quantities Quantities that have ONLY magnitude. • •E.g. Mass, volume, temperature, energy & etc. •Also known as vector quantities Quantities that have BOTH magnitude & direction. • •E.g. Velocity, acceleration, force, momentum & etc. Vectors are represented geometrically by directed line segments. Consider the vector such that . Then, is represented geometrically as a line segment directed from point to point as illustrate below: E.g. Vectors Representation 2 capital letter with arrow above , such as [Direction: point OR A vector may be written in a small letter with a curly line below it, such as [a bold small letter such as will be used when typing] Remarks: 5.1 Vector in Three Dimensions C05 - SM025 Page 1
Zero vector •Also known as Null vector •A vector such that its magnitude is 0 & has NO direction. Unit vector •A vector such that its magnitude is 1. the unit vector of E.g. Position vector A vector with the direction from the origin to a particular point • •Any arbitrary point can be written as a position vector. Consider is the origin and is an arbitary point, then is the position vector of point . E.g. Free vector A vector with specific magnitude and direction BUT has no fixed position. • in the vector equation of a line in the vector equation of a plane E.g. Equal vectors 2 vectors are equal vectors if both of them have the same magnitude and the same direction. • Negative vector 2 vectors are the negative vector of each other if both the vector has the same magnitude but opposite direction. • is the negative vector of [ If , then . ] E.g. Types of Vectors C05 - SM025 Page 2
Consider Addition & Subtraction of Vectors Scalar Multiplication of Vectors is parallel to & have same direction. & have opposite direction. Remarks: If where is a scalar , then are parallel. The magnitude of vector is denoted as and given by Magnitude of a Vector [Norm of a Vector] Remarks: The magnitude of vector is the length of the directed line segment which represents vector The unit magnitude of vector is denoted as and given by Unit Vector [vector with magnitude of 1 unit] Vector may also be written the following form: Vectors in Three Dimensions C05 - SM025 Page 3
Direction Cosines of a vector are the cosines of the angles between the vector and the three coordinate axes. • Consider point & as the origin . Let Diagram reference : https://www.geogebra.org/m/g8p4dqdj Direction cosines of Direction angles of axis axis axis Remarks: Direction Cosines of a Vector C05 - SM025 Page 4
Given and . Find the norm of . Solution : Find the unit vector of . Solution : Find the direction cosines of . Solution : Direction cosines of are Find the direction angles of of . Solution : Direction angles of are Find and . Solution : and Example 1 * C05 - SM025 Page 5
Example 2 ** Given and Find the values of and if is parallel to . Solution : Let By comparing component By comparing component By comparing component Example 3 ** Determine whether the vectors and are parallel. Solution : Let For component , For component , For component , Note that is NOT unique for all components , in . Therefore, and are NOT parallel. Example 2 & 3 ** C05 - SM025 Page 6