Worksheet 1.3: Introduction to the Dot and Cross Products
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1 Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot and Cross Products From the Toolbox (what you need from previous classes Trigonometry: Sine and cosine functions. Vectors: Know what a vector is. Be able to compute the magnitude of a vector. The dot and cross products are vector operations with many important applications. Today, you will focus on computation and geometric interpretations. In this worksheet, you will: Compute the dot and cross product of two vectors, using both the algebraic and geometric definitions. Definitions v = v 1, v 2, v 3 = v1 î + v 2 ĵ + v 3 ˆk w = w 1, w 2, w 3 = w1 î + w 2 ĵ + w 3 ˆk Dot Product Algebraic Definition: v w = v 1 w 1 + v 2 w 2 + v 3 w 3 Geometric Definition: v w = v w cos θ Cross Product Algebraic Definition: v w = w 1 w 2 w 3 Geometric Definition: Magnitude: Direction: v w = v w sin θ = area of the parallelogram spanned by v and w determined by the right-hand rule, and orthogonal to both v and w
2 Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot & Cross Products 1 Model 1: The Dot Product Algebraic and Geometric Definitions The dot product can be computed in two different ways: Algebraic Definition: v w = v 1 w 1 + v 2 w 2 Geometric Definition: v w = v w cos θ Diagram 1A: Vectors: r, s. Diagram 1B: Vectors a, b. Diagram 1C: Vectors v, w. Critical Thinking Questions In this section, you will practice computing the dot product, using both the algebraic and the geometric definitions.
3 Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot & Cross Products 2 (Q1 Only two out of the three diagrams 1A, 1B, 1C contain enough information for you to determine the component forms of their vectors. Which ones are they? (Q2 One of the diagrams out of the three 1A, 1B, 1C gives explicit information about the magnitudes of the vectors and the angle between them. Which one is it? (Q3 Compute the dot product r s for the vectors in Diagram 1A. Which definition did you use, and why did you choose that definition? (Q4 Compute the dot product a b for the vectors in Diagram 1B. Which definition did you use, and why did you choose that definition? (Q5 Compute the dot product F G for the vectors in Diagram 1C. Which definition did you use, and why did you choose that definition? (Q6 Now, compute the dot product F G for the vectors in Diagram 1C using the other definition of the dot product. How does this compare to your answer to (Q5? ( Q7 Suppose you know that two vectors v and w are perpendicular. What is the value of the dot product v w? ( Q8 What is the value of the dot product 0 w? ( 0 is the zero vector: the vector whose components all equal zero. ( Q9 Suppose you know that the angle between two vectors v and w is θ = π/3, and v w = 4. What (if anything can you say about the magnitudes v and w?
4 Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot & Cross Products 3 Model 2: The Cross Product Algebraic Definition Computing 2 2 determinants: DIAGRAM 2: a b c d = ad bc v w = = Algebraic Definition of the Cross Product: v = v 1 î + v 2 ĵ + v 3 ˆk w 1 w 2 w 3 w 1 w 2 w 3 î w 1 w 2 w 3 w = w 1 î + w 2 ĵ + w 3 ˆk ĵ + w 1 w 2 w 3 (1 ˆk (2 = v 2 v 3 w 2 w 3 î v 1 v 3 w 1 w 3 ĵ + v 1 v 2 w 1 w 2 ˆk (3 = ( ( (v 2 w 3 v 3 w 2 î ĵ + ˆk (4 Critical Thinking Questions In this section, you will practice computing the cross product using the algebraic definition. (Q10 In Diagram 2, look at the 3 3 array in front of the î basis vector in Equation (2. Draw a line through the top row and the first (left column. Compare the remaining values to the 2 2 array in front of the î basis vector in Equation (3 they should be the same. (Q11 Now, look at the 3 3 array in front of the ĵ basis vector in Equation (2. Draw a line through the top row and the second (middle column. Compare the remaining values to the 2 2 array in front of the ĵ basis vector in Equation (3 they should be the same. (Q12 Finally, look at the 3 3 array in front of the ˆk basis vector in Equation (2. Draw a line through the row and the column so that the remaining values are the same as in the 2 2 array in front of the ˆk basis vector in Equation (3. (Q13 Use the rule for computing 2 2 determinants at the top of Diagram 2 to complete Equation (4 in Diagram 2.
5 Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot & Cross Products 4 Equation (4 in Diagram 2 is the (algebraic definition of the cross product v w, but it s difficulty to memorize. You may find it is easier to remember how to compute v w if you begin with Equation (1, work through the process of crossing out rows and columns, then compute the three 2 2 determinants. (Q14 Compute v w for v = 3 î + ĵ 4 ˆk and w = 5 î + 2 ˆk = 5 î + 0 ĵ + 2 ˆk. (Q15 Using the same vectors v and w from (Q14, compute w v. (Q16 Complete this statement, based on your answers to (Q14 and (Q15: Statement: Since w v = v w, this means v w and w v have the magnitude, but directions. (Q17 Circle and correct the errors in the following equation (compare to Diagram 2, Equation (4: ( v w = (v 2 w 3 v 3 w 2 î + (v 1 w 3 v 3 w 1 ĵ + v 1 w 2 v 2 w 1 ˆk (Q18 Circle and correct the errors in the following equation : v w = (v 2 w 3 + v 3 w 2 î (v 1 w 3 + v 3 w 1 ĵ + ( v 1 w 2 + v 2 w 1 ˆk ( Q19 Show that 2, 4, 6 3, 1, 5 = 2 ( 1, 2, 3 3, 1, 5. Scalar multiples can be factored out of cross products: if v = cu, then v w = c(u w.
6 Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot & Cross Products 5 Interlude: The Right-Hand Rule We will go over this as a class: If you get to it early and want to begin working on it, here s a link to a video (by BraunVideos demonstrating the right-hand rule. (Video may auto-play. The right-hand rule is a method for determining how three vectors are positioned in space relative to each other. If a set of three vectors satisfies the right-hand rule, we will say that they are positively oriented. If not, we will say they are negatively oriented. There are several ways to remember the right-hand rule. Here is one: Start with three vectors {v, w.u} (the order of the three vectors matters!. Hold out your right hand so that your wrist is at the base point, your hand and fingers point in a straight line along v, and your palm faces w. Make a fist with your thumb sticking up (like you are hitch-hiking. If the third vector u points in the direction of your thumb, then u satisfies the right-had rule and {v, w.u} are positively oriented. The cross product of two vectors satisfies the right-hand rule: the set of vectors {v, w, v w} is positively oriented. Question: Since the two vectors v and w (pictured above are in the page, does their cross product v w point towards you (the viewer, or away from you (the viewer?
7 Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot & Cross Products 6 Model 3: The Cross Product Geometric Definition DIAGRAM 3: The cross product v w is a vector, so it has both magnitude and direction. If v = 0 or w = 0, then v w = 0. Otherwise: Geometric Definition of the Magnitude of v w (v, w 0: v w = v w sin θ Geometric Definition of the Direction of v w (v, w 0: v w is orthogonal to both v and w v w points in the direction determined by the right-hand rule. Critical Thinking Questions In this section, you will practice computing the magnitude and finding the direction of the cross product of two vectors using the geometric definitions. (Q20 On Diagram 3, sketch the parallelogram spanned by v and w, and the parallelogram spanned by u and v. Fact: Since v w sin θ is the area of the parallelogram spanned by v and w, then the magnitude of the cross product v w is the area of the parallelogram spanned by v and w. (Q21 In Diagram 3: Find the area of the parallelogram spanned by v and w.
8 Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot & Cross Products 7 (Q22 Complete the table, using the information from Diagram 3. When determining the direction of a cross product, assume that the paper is the plane containing u, v, and w, and the cross product points either towards you when looking at the diagram, or away from you when looking at the diagram. Cross Product Magnitude (Area of Parallelogram Direction (RHR v w v w = Towards / Away from the viewer w v w v = Towards / Away from the viewer v u v u = Towards / Away from the viewer u v u v = Towards / Away from the viewer (Q23 In Diagram 3: Use the given information to find the angle between u and w, and use this angle to compute u w. (Q24 Complete the statement: Statement: If two vectors a and b are parallel, then the angle between them is either θ = or θ =. So the magnitude of the cross product of a and b is either: ( ( a b = a b sin = or a b = a b sin =. The only vector with magnitude equal to zero is the zero vector. parallel, then: a b =. So, if a and b are ( Q25 Sketch two vectors v and w so that v w = 4 and v w points up out of the paper at the viewer.
9 Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot & Cross Products 8 Summary Dot Product Cross Product Output is a: Vector / Scalar Vector / Scalar Order of vectors matters: Yes / No Yes / No Trig function in the geometric definition is: sin θ / cos θ sin θ / cos θ The algebraic definition of the dot product of two vectors v = v 1, v 2, v 3 and w = w1, w 2, w 3 is v w =. The algebraic definition of the cross product of two vectors v = v 1 î + v 2 ĵ + v 3 ˆk and w = w 1 î + w 2 ĵ + w 3 ˆk is: v w = The geometric definition of the dot product of two vectors v and w is: v w =, where is the angle between v and w. The geometric definition of the magnitude of the cross product of two 3-d vectors v and w is: v w =, where is the angle between v and w. The direction of the cross product of two vectors v w is orthogonal to v and w, in the direction determined by the -hand rule. Computing the dot product using the algebraic definition will give the same / a different answer as using the geometric definition. Computing the cross product using the algebraic definition will give the same / a different answer as using the geometric definition. v w = w v, so changing the order of the vectors the direction of the cross product. If v and w are parallel, then v w =.
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