MTH 254 Study Guide - Midterm Guide: Unit Circle, Unit Vector, Cross Product

Carry out the following steps to determine the distance between the point P and the line L through the origin with direction v. Find a vector v = 1,b in the direction of line L. Find the position vector u corresponding to point P. Find projvu. Find w = u - projvu and |w|. Explain why |w| is the distance between P and L. P(2,-4); L is y = - x v = 1, u = (Type integers or simplified fractions.) projvu = (Type integers or simplified fractions.) d.w = (Type integers or simplified fractions.) |w| = (Type an exact answer, using radicals as needed.) Why is |w| the distance between the point and the line? The vector w is orthogonal to the line L. When its head is placed at P, its tail is at the origin. Therefore, its length is the same as the distance from the point to the origin, so it is the same as the distance from the point to the line. The vector w is orthogonal to the line L. Since the tail of the vector can be placed at the point P, then its length must be the distance from the point to the line. The vector w is orthogonal to the line L. When its head is placed at P, its tail is on the line. Therefore, its length is the same as that of the perpendicular line segment connecting the point to the line.

Determine a and b if A(2, -3, 4), B(11, -9, 7) and C(-13, a, b) are collinear. Find a vector v in the opposite direction to w = (-1, 4, 1) and with a magnitude of 6 units. Find scalars a and b such that a(1, -1) + b(2, 5) = (-8, -27). Determine the angle between the lines x - y = 3 and 3x + 2y = 11. Determine c if 2i + cj + (c - 2)k and ci + 3j + ck are orthogonal Determine all vectors orthogonal to both (2, 3, -1) and (1, -2, 2). An aeroplane needs to fly due east from one city to another at a speed of 400 km/h. If a 50 km/h wind is blowing from the north-east, in which direction must the plane head to compensate for the wind, and what is the speed of the plane in still air? BC is parallel to OA and is twice its length. Find, in terms of p and q, vector expressions for AC and OM.

The points P, Q, R, and S, joined by the vectors u, v, w, x, are the vertices of a quadrilateral in R3. The four points needn't lie in a plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system. Use vector addition to show that u + v = w + x. Choose the correct answer below. u + v = w + x Let m be the vector that joints the midpoints of PQ and QR. Show that m = (u + v)/2. Choose the correct answer below. Let n be the vector that joints the midpoints of PS and SR. Show that n = (x + w)/2. Choose the correct answer below. Combine parts (a), (b), and (c) to conclude that m = n. Since both u + v and w + x are equal to , m and n are both equal to , and so m = n. Since both u + v and w + x are equal to , m and n are both equal to 1/2 , and so m = n. Since both u + v and w + x are equal to , m and n are both equal to , and so m = n. Since both u + v and w + x are equal to , m and n are both equal to 1/2 , and so m = n. Explain why part (d) implies that the lines segments joining the midpoints of the sides of the quadrilateral form a parallelogram. Using a similar process, it can be shown that the side joining the tails and the side joining the heads of m and n (denoted by dashed lines in the figure) also have the same length and direction, forming a parallelogram. Since m = n, u and w are parallel, and x and v are parallel. It follows that each pair of parallel vectors have equal length, forming a parallelogram. Since m and n have the same direction, they are parallel. Since m and n are parallel and have the same length, the side joining the tails and the side joining the heads of m n (denoted by dashed lines in the figure) must also be of equal length and be parallel, forming a parallelogram.