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Spring 2015 Math 152 Quiz 9 Solutions Wed, 22/Apr/2015 1. Find the two unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4). • The slope of the tangent line at the point is dy 4 rise dx x=2 = 2x x=2 = 4 = 1 = run . • A vector parallel to the tangent line is v = [1, 4]. • Two unit vectors parallel to thei tangent line are h [1,4] v 1 u = kvk = √1+16 = √17 , √417 and h i −u = − √117 , − √417 . 2. Find an equation of the sphere with center C (2, −6, 4) and radius r = 5. Then describe its intersection with each of the coordinate planes x = 0 (yz-plane), y = 0 (xz-plane), z = 0 (xy-plane), both geometrically (in English) and analytically (equations). [Some of the intersections may be the empty set, 0.] / • The sphere has equation (x − 2)2 + (y + 6)2 + (z − 4)2 = 25. • When x = 0 is substituted into the sphere’s equation, we have (y + 6)2 + (z − 4)2 = 21, x = 0. Hence the sphere intersects the yz-plane in the √ circle with center C (0, −6, 4) and radius r = 21. • When y = 0 is substituted into the sphere’s equation, we have (x − 2)2 + (z − 4)2 = −11, a contradiction, since a sum of squares of real numbers is never negative. Hence the sphere does not intersect the xz-plane. In other words, the intersection is the empty set 0. / • When z = 0 is substituted into the sphere’s equation, we have(x − 2)2 + (y + 6)2 = 9, z = 0. Hence the sphere intersects the xy-plane in the circle with center C (2, −6, 0) and radius r = 3. 3. Find the three angles of the triangle with vertices A (1, 0, −1), B (3, −2, 0), C (1, 3, 3), in degrees. • Label the angles at vertices A, B, C as α, β , γ, respectively. Manipulate the Physics definition of dot product: v·w . v · w = kvk kwk cos θ =⇒ θ = cos−1 kvkkwk ! • So α = cos−1 − →− → AB· − AC → →− ABAC ≈ 97.66◦ . Similarly, β ≈ 53.50◦ and γ ≈ 28.84◦ ; α + β + γ = 180◦ . 4. Find the work done by a force F = 8i − 6j + 9k that moves an object from the point P (0, 10, 8) to the point Q (6, 12, 20) along a straight line. Here distance is measured in meters and the force in newtons. Include correct units of work. • The displacement vector is D = Q − P = [6, 2, 12]. • The work done is W = F · D = 48 − 12 + 108 = 144 joules. 5. Use vectors to determine whether the triangle with vertices P (1, −3, −2), Q (2, 0, −4), and R (6, −2, −5) is right-angled. • Form three vectors which represent the sides of the triangle. u = Q − P = [1, 3, −2] v = R − Q = [4, −2, −1] w = P − R = [−5, −1, 3] • Compute the lengths of these vectors. √ √ √ kuk = 14, kvk = 21, kwk = 35 • Since 14 + 21 = 35, these lengths satisfy the Pythagorean Theorem. Hence the triangle is right-angled! 6. [BONUS] Find values ofx such that the angle 2 1 −1 between the vectors v = and ◦ w = 1 x 0 is 45 . Give exact values for x and approximations to 2 decimal places. • Start with the Physics definition of dot product and see where the road leads. v · w = kvk kwk cos 45◦ √ p 1 2+x = 6 1 + x2 √ 2 2 2 4 + 4x + x = 3 1 + x 0 = 2x2 − 4x − 1 √ 4 ± 16 + 8 x = 4 √ 1 x = 1± 2 6 x ≈ −0.22, 2.22 • “All roads lead to Rome!”