Quantities that involve BOTH a magnitude and a direction are called vectors. Quantities that involve magnitude, but no direction are called scalars.

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1 Math 1330 Vectors Quantities that involve BOTH a magnitude and a direction are called vectors. Quantities that involve magnitude, but no direction are called scalars. Example: You are driving south at 55 miles per hour. The magnitude is the speed of your car and the direction is south. Geometrically, a vector is a directed line segment, i.e. a line segment that has an initial point and a terminal point. Equal vectors have the same magnitude and the same direction. Example: Show that a vector uu with initial point ( 4, 1) and a terminal point ( 1, 2) and a vector vv with initial point (1,1) and terminal point (4,4) are equal. Slope of uu = yy 2 yy 1 xx 2 xx 1 Slope of vv = yy 2 yy 1 xx 2 xx 1 Magnitude of uu = uu = (xx 2 xx 1 ) 2 + (yy 2 yy 1 ) 2 Magnitude of vv = vv = (xx 2 xx 1 ) 2 + (yy 2 yy 1 ) 2 Page 1 of 8

2 Scalar multiplication A vector can be multiplied by a real number. Multiplying a vector by any positive real number other than 1 changes its magnitude, but not its direction. Multiplying a vector by any negative real number other than -1 changes its magnitude, AND its direction. If kk is a real number and vv, the vector kkvv is called a scalar multiple of the vector vv. The magnitude of kkvv is kk vv. The direction of kkvv is the same as vv if kk > 0, and opposite the direction of vv if kk < 0. Vector Addition (Geometric Method) To find the sum, uu + vv, of vectors uu and vv : 1. Position uu and vv, so that the terminal point of uu coincides with the initial point of vv. 2. The resultant vector, uu + vv, extends from the initial point of uu to the terminal point of vv. Page 2 of 8

3 The difference of two vectors, uu vv, is defined as uu + ( vv ). Unit Vectors and Representing Vectors in Rectangular Coordinates Vector ıı is the unit vector whose direction is along the positive x-axis. Vector ȷȷ is the unit vector whose direction is along the positive y-axis. Vectors in the rectangular coordinate system can be represented in terms of ıı and ȷȷ. Vector vv, from (0,0) to (aa, bb), is represented as vv = aaıı + bbȷȷ, where aa is the horizontal component of vv, and bb is the vertical component of vv. The magnitude of vv = aaıı + bbȷȷ is equal to aa 2 + bb 2. aaıı + bbȷȷ is called a linear combination of the vectors ıı and ȷȷ. Example: Sketch the vectors vv = 4ıı + 3ȷȷ and uu = 2ıı 3ȷȷ and find their magnitudes. Page 3 of 8

4 Vector vv with initial point PP 1 = (xx 1, yy 1 ) and terminal point PP 2 = (xx 2, yy 2 ) is equal to the position vector vv = (xx 2 xx 1 )ıı + (yy 2 yy 1 )ȷȷ. Example: Let vv be a vector from initial point ( 8,6) to terminal point ( 2, 3). Write vv in terms of ıı and ȷȷ. Example: Let vv be a vector from initial point ( 3,4) to terminal point (6, 4). Find the vertical and the horizontal component of this vector. Vertical component: Horizontal component: Operations with Vectors If vectors are expressed in terms of ıı and ȷȷ, we can easily carry out operations such as vector addition, vector subtraction, and scalar multiplication. If uu = aaıı + bbȷȷ and vv = ccıı + ddȷȷ, then uu + vv = (aa + cc)ıı + (bb + dd)ȷȷ uu vv = (aa cc)ıı + (bb dd)ȷȷ kkuu = (kkkk)ıı + (kkkk)ȷȷ Example: If uu = 2ıı + 3ȷȷ and vv = ıı 4ȷȷ, find a. uu + vv Page 4 of 8

5 Example cont.: If uu = 2ıı + 3ȷȷ and vv = ıı 4ȷȷ, find b. uu vv c. 3uu 2vv Unit Vectors A unit vector is a vector whose magnitude is one. In many applications it is useful to find the unit vector that has the same direction as a given vector. For any nonzero vector vv, the vector vv vv is the unit vector that has the same direction as vv. To find this vector, divide vv by its magnitude. Example: Find the unit vector in the same direction as vv = 4ıı + 3ȷȷ. Verify that the vector has magnitude 1. Page 5 of 8

6 A Vector in Terms of Its Magnitude and Direction The components aa and bb of the vector vv = aaıı + bbȷȷ can be expressed in terms of the magnitude of vv and the angle θθ it makes with the positive x-axis. vv = aaıı + bbȷȷ = vv cos θθ ıı + vv sin θθ ȷȷ θθ is called the direction angle. Example: Write the vector vv in terms of ıı and ȷȷ if its magnitude and direction angle are given. vv = 5, θθ = 60. Example: Given the vector vv = 3ıı + 3 3ȷȷ, find the direction angle. Page 6 of 8

7 The Dot Product of Two Vectors If uu = aaıı + bbȷȷ and vv = ccıı + ddȷȷ, then the dot product of uu vv = aaaa + bbbb. The dot product of two vectors is the sum of the products of their horizontal components and vertical components. The result of vector addition and scalar multiplication is a vector. The result of the dot product is a SCALAR (a real number). Example: If uu = 2ıı + 3ȷȷ and vv = ıı 4ȷȷ, find uu vv, vv uu, and vv vv. Properties of Dot Product 1. uu vv = vv uu 2. 0 vv = 0 3. vv vv = vv 4. uu (vv + ww ) = uu vv + uu ww 5. (cuu ) vv = cc(uu vv ) = uu (ccvv ) Alternative Formula for the Dot Product: uu vv = uu vv cos θθ, where θθ is the smallest non-negative angle between vectors uu and vv. Formula for the Angle between Two Vectors: cos θθ = uu vv and θθ = uu vv cos 1 uu vv. uu vv Example: If uu = 3ıı + 4ȷȷ and vv = 5ıı + 12ȷȷ, find the angle between uu and vv. Page 7 of 8

8 Parallel and Orthogonal Vectors Two vectors are parallel when the angle θθ between the vectors is 0 or 180. If θθ = 0, the vectors point in the same direction. If θθ = 180, the vectors point in opposite directions. Two vectors are orthogonal when the angle between them is 90. Two nonzero vectors are orthogonal if and only if uu vv = 0. Example: Are the vectors uu = 2ıı + 3ȷȷ and vv = 12ıı 8ȷȷ orthogonal? Example: Determine if uu = 3ıı 5ȷȷ and vv = 6ıı 10ȷȷ are parallel, orthogonal, or neither. Example: Given vectors 2,3 and 4, xx are orthogonal, determine the value of xx. Page 8 of 8