Wed. Apr. 15– Calculus Lecture #23.2 Vectors, Dot Products, Projections, and Planes 11.1.9) If v and w are unit vectors, what is the geometrical meaning.

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Wed. Apr. 15– Calculus Lecture #23.2 Vectors, Dot Products, Projections, and Planes ) If v and w are unit vectors, what is the geometrical meaning of v·w? What is the geometrical meaning of (v·w)w? Draw a figure with v = i and w = (3/5)i +(4/5)j.

(based on ) A = (1, –1, 0), B = (4, 2, 4). Find the projection P of B along (parallel to) A. Find the projection of B perpendicular to A.

(a) Draw the unit vectors u 1 = (cos , sin  ) and u 2 = (cos , sin  ). By dot products, find the formula for cos(  –  ).

) A methane molecule has a carbon atom at (0, 0, 0) and hydrogen atoms at (1, 1, –1), (1, –1, 1), (–1, 1, 1), and (–1, –1, –1). Find (a) the distance between the hydrogen atoms (b) the angle between the vectors going out from the carbon atom to the hydrogen atom.

11.2.8) Find the equation of the plane through P 0 = (1, 2, –1) perpendicular to N = i + 2j – k ) Find two points P and P 0 on the plane x + 2y + 3z = 6 and a normal vector N. Verify that N·(P – P 0 ) = 0.

) From their equations, how can you tell if two planes are (a) parallel (b) perpendicular (c) at a 45 o angle? use planes ax + by + cz = d Ex + Fy + Gz = H ) Find the equation of the plane through (1, 0, 1) parallel to x + 2y + z = 0. (based on ) What is the distance between the planes x + y + 5z = 7 and 3x + 2y + z = 1?

) How far is the plane x + y – z = 1 from (0, 0, 0) and also from (1, 1, –1)? Find the nearest points.

) In the semicircle find C and D in terms of A and B. Prove that C·D = 0 (they meet at right angles).