Worksheet 1.4: Geometry of the Dot and Cross Products
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1 Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products From the Toolbox (what you need from previous classes): Basic algebra and trigonometry: be able to solve quadratic equations, know the cosine and sine, and inverse cosine functions. Vectors: Angle-bracket and î, ĵ, ˆk notation; computing vector magnitude; computing the dot and cross products of two vectors. The dot product is used to measure angles between vectors. The cross product is used to measure the area of parallelograms spanned by two vectors, and to produce a vector normal to a plane containing two vectors. In this worksheet, you will: Use the dot product to measure angles between vectors, with a special focus on perpendicular (orthogonal) vectors. Use the cross product to measure the area of parallelograms and triangles, and to find normal vectors to planes. Definitions Dot Product (Geometric Definition): v w = v w cos θ Cross Product (Geometric Definition): Magnitude: v w = v w sin θ = area of the parallelogram spanned by v and w Direction determined by the right-hand rule, and orthogonal to both v and w
2 Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products 1 Model 1: The Dot Product & Angles Diagram 1A: a = 6 b = 3 a b = 9 Diagram 1B: Diagram 1C: u = 1, 1, 1 v = 1, 1, 2 Geometric Definition of the Dot Product: v w = v w cos θ Critical Thinking Questions In this section, you will use the dot product to compute angles between vectors. (Q1) In Diagram 1A: What are the magnitudes of the vectors a and b? a = b =
3 Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products 2 (Q2) In Diagram 1A: What is the value of the dot product a b? a b = (Q3) Use the geometric definition of the dot product and your answers from (Q1) and (Q2) to find cos θ, where θ is the angle between a and b in Diagram 1A. (Simplify your answer.) cos θ = (Q4) Using your answer from (Q3), find the angle θ between a and b in Diagram 1A. Answer in both degrees and in radians. (Give exact answers for both.) θ = or θ = (radians) (Q5) Using the same ideas as in (Q3) and (Q4), write a general formula that can be used for computing the angle θ between the two arbitrary vectors v and w in Diagram 1B. ( Arbitrary vectors means you don t have an explicit algebraic form).) Formula: θ = (Q6) In Diagram 1C: Sketch and label the x-, y-, and z-coordinate axes (3-space), then add the vectors u = 1, 1, 1 and v = 1, 1, 2 to your sketch. ( Q7) In Diagram 1C: Use your sketch to estimate the angle θ between u and v, using degrees (we use radians for derivatives and integrals, but degrees are usually easier to visualize). How confident are you that your estimate is a good one? ( Q8) Now, use the formula from (Q5) to compute the angle θ between the vectors u and v in Diagram 1C. Compare this to your answer in (Q7): how good was your estimate? ( Q9) For the triangle that has sides u and v in Diagram 1C, find the measures of all three angles.
4 Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products 3 Model 2: The Dot Product & Orthogonality Diagram 2A: î = 1, 0, 0 ĵ = 0, 1, 0 ˆk = 0, 0, 1 Pair of Vectors Angle Between Vectors Dot Product of Vectors î, ĵ θ î, ĵ = 90 î ĵ = î, ˆk θ î, ˆk = î ˆk = 1, 0, 0 0, 0, 1 = 0 ĵ, ˆk θ ĵ, ˆk = ĵ ˆk = Diagram 2B: u = 0, 2 v = 1, 1 w = 2, 2 Pair of Vectors Angle Between Vectors Dot Product of Vectors u, v θ u,v = 45 u v = u, w θ u,w = u w = 0, 2 2, 2 = 4 v, w θ v,w = v w =
5 Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products 4 Critical Thinking Questions In this section, you will use the dot product to determine when two vectors are orthogonal. (Q10) In Diagram 2A: The angles between the three pairs of vectors are all the same. Write down this angle on the table in Diagram 2A. (Q11) In Diagram 2A: The dot products between the three pairs of vectors are all the same. Write down this dot product on the table in Diagram 2A. (Q12) In Diagram 2B: Sketch and label the vectors u = 0, 2, v = 1, 1, w = 2, 2 on the coordinate axes, and complete the table. (Q13) From the table in Diagram 2B: Which vectors are perpendicular? (Q14) From the table in Diagram 2B: Which vectors have a dot product equal to zero? (Q15) Based on the results of questions (Q10-14), complete the following statement. Statement: If two vectors a and b are perpendicular, then a b =. (Q16) Use the geometric definition of the dot product, a b = a b cos θ, to explain why the statement in (Q15) is true for all pairs of perpendicular vectors. Definition: Two vectors v and w are orthogonal if v w = 0. (Q17) For every pair in the set of vectors: v = 1, 1, 2, w = 1, 1, 1, u = 3, 1, 1, 0 = 0, 0, 0 compute the dot product to determine which pairs are orthogonal. Dot Product: v w = v u = v 0 = w u = w 0 = u 0 = Orthogonal? Y / N Y / N Y / N Y / N Y / N Y / N (Q18) The following statement is true, but incomplete. Complete the statement. (Hint: Look at the geometric definition of the dot product, and (Q17).) Statement: If two vectors are orthogonal, then they are perpendicular (the angle between them is 90 ), or...
6 Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products 5 ( Q19) For a = a 1, a 2, a 3 and b = b1, b 2, b 3, show that a b is orthogonal to a by computing (a b) a. Model 3: The Cross Product Diagram 3A: v w = 6 ˆk v w = 6 Diagram 3B: a b = 2 î ĵ + 5 ˆk a b = 30 Critical Thinking Questions In this section, you will use the cross product to measure the area, and to find vectors normal to planes. (Q20) In Diagrams 3A and 3B, sketch the parallelograms spanned by the pairs of vectors v and w, and a and b.
7 Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products 6 Recall: The magnitude of the cross product of two vectors is the area of the parallelogram spanned by the two vectors. (Q21) What are the areas of the parallelograms you sketched in (Q20)? Area of parallelogram spanned by v and w (Diagram 3A): Area of parallelogram spanned by a and b (Diagram 3B): (Q22) In Diagrams 3A and 3B, sketch the triangles that have two sides formed by the pairs of vectors v and w, and a and b. What are the areas of these triangles? Area of triangle with sides v and w (Diagram 3A): Area of triangle with sides a and b (Diagram 3B): Definition: A vector that is perpendicular to a plane is called a normal vector to the plane. It is orthogonal to every vector in the plane. Two non-parallel, non-zero vectors v and w determine a plane (in the same way that three non-collinear points determine a plane). The cross product v w is orthogonal to both v and w, so it is normal to the plane determined by v and w. (Q23) In Diagram 3A: The xy-plane contains the vectors v and w, so the cross product v w is normal to the xy-plane. The unit vector in the direction of v w is. (Q24) In Diagram 3B: The vector a b is normal to the plane P containing a and b. Sketch a vector that gives the direction of a b (don t worry about the magnitude, just the direction). ( Q25) Every plane in R 3 has two unit normal vectors (vectors that are perpendicular to the plane, with magnitude equal to one). Find the two unit normal vectors to the plane P in Diagram 3B. ( Q26) Which of the two vectors R = 4 î 2 ĵ + 10 ˆk and S = 2 î ĵ + 7 ˆk is normal to the plane P in Diagram 3B? Explain how you determined this.
8 Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products 7 ( Q27) Sketch two vectors F and G so that F G = 10 ˆk. Summary The dot product can be used to find the between two vectors. If v and w are perpendicular, then v w =. If v w = 0, we say that v and w are. If v and w are orthogonal, then either: The angle between them is θ =, or: v = or w =. The cross product of two vectors is to both of the original vectors. This means the cross product of two (non-zero, non-parallel) vectors is to the plane containing the vectors. The magnitude v w is the of the spanned by v and w. To determine whether two vectors are perpendicular, use the. To find a vector that is perpendicular to two vectors, use the. To find the area of a parallelogram or triangle, use the.
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