Q1 The position vectors of the points A and B are a and b respectively. The point C lies on AB such that AC: CB = m: n. Prove that the position vector the point C is 1 / (m + n) (mb + na). (a) The position vectors of the points P and Q are given by OP = 3a - 2b and OQ = a + b. The line joining P and Q is divided internally and externally into the ratio 2: 1 by the points R and S respectively. Find the position vectors of points R and S in terms of a and b. (b) Consider the vectors AE = 3a - 2b and DC = 2a + 4b. E and B are the mid points of AD and AC respectively. Find an expression for EB in terms of a and b. Does there exist a and b such that EB is parallel to DC? Justify your answer. (c) OPQ is a triangle such that OP = p and OQ = q. M lies on the line PQ such that PM: MQ = 1: 3 and N lies on the line OQ such that ON: NQ = 1: 1. If OM and PN intersect at L prove that OL = 1/5 (3p + q).
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Q1. Prove that the points with vectors a+2b, 2a-b, and 3a-4b are collinear. Q2. E and F are points on the sides AD and BC of a quadrilateral ABCD such that AE=ED and BF=FC. If P, Q, and R are the midpoints of AB, EF, and DC respectively, show that P, Q, and R are collinear and that PQ=QR.
Madhur L.
The points $P, Q, R,$ and $S,$ joined by the vectors $\mathbf{u}, \mathbf{v}, \mathbf{w},$ and $\mathbf{x},$ are the vertices of a quadrilateral in $\mathbb{R}^{3}$. 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. a. Use vector addition to show that $\mathbf{u}+\mathbf{v}=\mathbf{w}+\mathbf{x}$ b. Let $\mathbf{m}$ be the vector that joins the midpoints of $P Q$ and $Q R$ Show that $\mathbf{m}=(\mathbf{u}+\mathbf{v}) / 2$ c. Let $n$ be the vector that joins the midpoints of $P S$ and $S R$. Show that $\mathbf{n}=(\mathbf{x}+\mathbf{w}) / 2$ d. Combine parts (a), (b), and (c) to conclude that $\mathbf{m}=\mathbf{n}$ e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.
Vectors and Vector-Valued Functions
Vectors in Three Dimensions
In contrast to the proof in Exercise $81,$ we now use coordinates and position vectors to prove the same result. Without loss of generality, let $P\left(x_{1}, y_{1}, 0\right)$ and $Q\left(x_{2}, y_{2}, 0\right)$ be two points in the $x y$ -plane and let $R\left(x_{3}, y_{3}, z_{3}\right)$ be a third point, such that $P, Q,$ and $R$ do not lie on a line. Consider $\triangle P Q R$ a. Let $M_{1}$ be the midpoint of the side $P Q$. Find the coordinates of $M_{1}$ and the components of the vector $\overrightarrow{R M}$ b. Find the vector $\overrightarrow{O Z}_{1}$ from the origin to the point $Z_{1}$ two-thirds of the way along $\overrightarrow{R M}_{1}$. c. Repeat the calculation of part (b) with the midpoint $M_{2}$ of $R Q$ and the vector $\overrightarrow{P M}_{2}$ to obtain the vector $\overrightarrow{O Z}_{2}$ d. Repeat the calculation of part (b) with the midpoint $M_{3}$ of $P R$ and the vector $\overline{Q M}_{3}$ to obtain the vector $\overrightarrow{O Z}_{3}$ e. Conclude that the medians of $\triangle P Q R$ intersect at a point. Give the coordinates of the point. f. With $P(2,4,0), Q(4,1,0),$ and $R(6,3,4),$ find the point at which the medians of $\triangle P Q R$ intersect.
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