Solution: 2. Sketch the graph of 2 given the vectors and shown below.

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1 7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit of measure such as square feet, cubic feet, miles, degrees, or miles per hour. A quantity of this type is a scalar quantity, and the corresponding real number is a scalar. Concepts such as velocity, acceleration, force, momentum, and electric field have both magnitude and direction and are often represented by a directed line segment. Another name for a directed line segment is a vector. To find the sum of two vectors and, we place the initial point of vector at the terminal point of vector. The sum +, is the vector with the same initial point as and the same terminal point as. 1. Sketch the graph of + and given the vectors and shown below. Solution: + Another method to sketch the graph of the sum of two vectors is to use the parallelogram rule. Place vectors and so that their initial points coincide. Then, complete a parallelogram that has and as two sides. The diagonal of the parallelogram with the same initial point as and is the sum Sketch the graph of 2 given the vectors and shown below. 2 2

2 Algebraic Interpretation of Vectors. A vector with its initial point at the origin is called a position vector. A position vector with its endpoint at the point, is written as,. The numbers a and b are the horizontal and vertical components of the vector. The positive angle between the x-axis and a position vector is called vector s direction angle. (a,b), Magnitude and Direction Angle of a Vector, The magnitude (length) of vector, is given by The direction angle satisfies tan In examples 3 and 4, find the magnitude and direction angle for each vector , 8 Solution: The magnitude is The direction angle satisfies tan so tan Since the position vector is in the 4 th quadrant, the direction angle is ,24 Solution: The magnitude is The direction angle satisfies tan so tan Since the position vector is in the 2 nd quadrant, the direction angle is Horizontal and Vertical Components The horizontal and vertical components, respectively, of a vector having magnitude and direction angle are given by cos and sin That is, cos, sin

3 5. Write the vector in the form, Solution:, cos, sin 8 cos 220, 8 sin , 5.14 Properties of Parallelograms A parallelogram is a quadrilateral whose opposite sides are parallel The opposite sides and opposite angles of a parallelogram are equal, and adjacent angles of a parallelogram are supplementary. The diagonals of a parallelogram bisect each other, but do not necessarily bisect the angles of the parallelogram. 6. Two forces of magnitude 116 lb and 139 lb act a point in the plane. The angle between the two forces is Find the magnitude of the resultant vector. Solution: Use the parallelogram rule for adding vectors Adjacent angles of a parallelogram are supplementary, therefore the angle between the two sides of the parallelogram is By the law of cosines 2cos cos cos

4 7. Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. 85 lb lb Solution : The angle between the two sides of the parallelogram is the supplement of the angle between the vectors, that is By the law of cosines: 2cos cos cos The Unit Basis Vectors The unit basis vectors are defined as: 1,0 0,1 Any position vector can be written as a linear combination of the unit basis vectors.,, 0 0, 1,0 0,1 In examples 8 and 9, write each position vector in the form 8. 7, 5 Solution: ,0 Solution: 4

5 Dot Product The dot product of the two vectors, and, is denoted by and defined as In problems 10-12, find the dot product for each pair of vectors , 2, 3,5 Solution: 6, 2 3, ,8, 3, 5 Solution: 3,8 3, , 3 2 Solution: Geometric Interpretation of the Dot Product If is the angle between two nonzero vectors, then 13. Find the angle between 5,2 6,3 Solution: = cos Theorem: If 0 then the vectors are orthogonal (perpendicular). 14. Determine if 1,2 6,3 are orthogonal. Solution: , thus the vectors are orthogonal.