(10/14/08)Math 10C. Lecture Examples.Section 12.3. The dot product†Example 1 Calculate v · w for v = h6 , −2i and w = h4, 3i.Answer: v · w = 18.Example 2 What is v · w for v = h6, −2, 3i and w = h4, 3, −6i?Answer: v · w = 0Example 3 Find an angle θ between the vectors v = h4, 1 i and w = h2, 4i in Figure 1.Give exact and approximate decimal values.x2 4y24v = h4, 1iw = h2, 4iθFIGURE 1Answer: θ = cos–112√17√20.= 0.862 radiansExample 4 Find the constant k such that the vectors h−3, −1i and hk, −2i areperpendicular. T hen draw the two vectors.Answer: k =23• The vectors are h23, −2i and h−3, −1i. • Figure A4x2y2−2h−3, −1ih23, −2i23Figure A4†Lecture notes to accompany Section 12.3 of Calculus by Hughes-Hallett et al.1Math 10C. Lecture Examples. (10/14/08) Section 12.3, p. 2Example 5 Give an equation of the plane through the point (2, 3, 4) and perpendicularto the vecto r h−6, 5, −4i.Answer: −6(x − 2) + 5(y − 3) − 4(z − 4) = 0Example 6 Give an equation of the plane through the point (1, −1, 2) that is parallel tothe plane 3x − 5y + 6z = 10.Answer: 3(x − 1) − 5(y + 1) + 6(z − 2) = 0 or (written) 3x − 5y + 6z = 20.Interactive ExamplesWork the following Interactive Examples on Shen k’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 12.2: Examples 1 and 2Section 12.5: Examples 5 and 6‡The chapter and section numbers on Shenk’s web site refer to his calculus ma nuscript and not to the chapters and sectionsof the textbook for the
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