2.8 &Vector components for a crane-boom. (Section 1.4) Shown right is a crane whose cab A supports a boom B that swings a wrecking ball Co i Jare right-handed orthogonal unit vectors with i horizontally-right, j vertically- upward, and k perpendicular to the plane con- taining points Ng. Ap. Be.Co Draw each position vector listed below and then use your knowledge of sine/cosine to re- solve these vectors into i and j components. A’s position vector fromm N AN Be’s position vector from Ap Re/Ap- C,’s position vector from Bee Be’s position vector from N, FRI N,’s position vector from C FN 2.9 Dot products and distance calculations. Show work (Section 2.10) Shown right is a crane whose cab A sup- ports a boom B that swings a wrecking ball Co. To prevent the wrecking ball from hitting a car at point No. the distance be- tween No and the tip of the boom (point y Be) must be controlled. To start this problem, express Bc’s posi- tion vector from No in terms of x. LB, and the unit vectors n, and b (a) Without resolving f Be/No into n, and n, components (done in the next step), use equation (2.4) and the distributive property to calculate the distance between No and Be in terms of z. Ln 0n· Then calculate its numerical value when r-20 m, LB-10 m. ?? = Result: (If necessary, conuplete the footnote hint below) Distance betweeu N, and Be: FBeN – 29.1 nm (b) Two colleagues are confused by your use of mized-bases vectors (ie,.i + Lgb,), and ask you to verify B’s position vector from No can be expressed in the uniform-basis as shown below. Use this uniform-basis expression to verify your previous result for jehe/N Note: This inelicient uniform-basis approach requires the simplifying trigosometric ileatity sin(0a) + co(0a)-1. Result: rBe/x,-Ir +11, cos(0B)]h, +Ly sin(@p)i, lise / N.1 sinpasm to previous msull. (c) Optional: Calculate the distance between No and Co in ternus of r, Lp, Lc. 0B. and 8c Hist: Use the distributive property to express FDe/N. Fe/N in terms of a. Lm, and i. b, Next, “se the dot-product definition in equatica’ (22) to calculate u..6. “ and theu rewrite Note The ditribeative property for vector dus-multiplication b(6+6) (+d)-aead+bebd 167 Homework 2 Vertrs with as
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