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 +. 2. Sketch the graph of 2 given the vectors and shown below. 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. 3. 15, 8 Solution: The magnitude is 15 8 289 17 The direction angle satisfies tan so tan 28.07. Since the position vector is in the 4 th quadrant, the direction angle is 360 28.07 331.93. 4. 7,24 Solution: The magnitude is 7 24 25 The direction angle satisfies tan so tan 73.74. Since the position vector is in the 2 nd quadrant, the direction angle is 180 73.74 106.26. 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
5. Write the vector in the form,. 8 220 Solution:, cos, sin 8 cos 220, 8 sin 220 6.13, 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 140 0 50. Find the magnitude of the resultant vector. Solution: Use the parallelogram rule for adding vectors. 139 116 Adjacent angles of a parallelogram are supplementary, therefore the angle between the two sides of the parallelogram is 180 140 50 39 10 By the law of cosines 2cos 116 139 2 116139 cos 39 10 116 139 2 116139 cos 39 10 88.17
7. Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. 85 lb 50 0 190 lb Solution : The angle between the two sides of the parallelogram is the supplement of the angle between the vectors, that is 130 0. By the law of cosines: 2cos 85 190 2 85190 cos 130 85 190 2 85190 cos 130 253.15 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: 7 5 9. 4,0 Solution: 4
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. 10. 6, 2, 3,5 Solution: 6, 2 3,5 6 3 2 5 18 10 8 11. 3,8, 3, 5 Solution: 3,8 3, 5 3 3 85 9 40 49 12. 5 12, 3 2 Solution: 5 12 3 25 3 12 2 15 24 9 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: 56 2 3 5 2 6 3 = 24 1305 cos 24 131.6 1305 Theorem: If 0 then the vectors are orthogonal (perpendicular). 14. Determine if 1,2 6,3 are orthogonal. Solution: 16 2 3 6 6 0, thus the vectors are orthogonal.