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B Week1Pset

The document contains 11 problems related to analytic geometry: 1) Find the unit vector of a vector between two points 2) Show that a point is on a line determined by two other points if its position vector is a linear combination of the position vectors of the two points 3) Prove a formula for the position vector of a point dividing a line segment between two other points in a given ratio 4) Label and sketch points determined by linear combinations of two vectors 5) Find values of x satisfying equations relating position vectors of points 6) Determine if sets of points are collinear 7) Find coordinates of a point given the midpoint of a segment between it and another point 8) Find mid

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Elena Iulia
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39 views1 page

B Week1Pset

The document contains 11 problems related to analytic geometry: 1) Find the unit vector of a vector between two points 2) Show that a point is on a line determined by two other points if its position vector is a linear combination of the position vectors of the two points 3) Prove a formula for the position vector of a point dividing a line segment between two other points in a given ratio 4) Label and sketch points determined by linear combinations of two vectors 5) Find values of x satisfying equations relating position vectors of points 6) Determine if sets of points are collinear 7) Find coordinates of a point given the midpoint of a segment between it and another point 8) Find mid

Uploaded by

Elena Iulia
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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ANALYTIC GEOMETRY, PROBLEM SET 1

1. Find the unit vector that has the same direction as vector v that begins at (1, 4, 10) and
ends at (3, 0, 4).
2. (The vector form of the equation of a line determined by two points) Let A and B be
points whose position vectors relative to an origin O of a Euclidean space are a and b. Show
that a point R is on the line determined by A and B if and only if there are λ, µ ∈ R such
that λ + µ = 1 the position vector of R is
r = λa + µb.

3. (The one and only Theorem) Let A and B be points whose position vectors relative to
an origin O of a Euclidean space are a and b. Let r be the position vector of a point R on
AB which divides [AB] in the ratio mn
. Prove that
na + mb
r= .
m+n
4. Let v = (1, 1) and w = (2, 0) in R2 .
(a) Label the nine points av + bw where a and b are one of 0, 1, 2.
(b) Sketch the points av + bw where a + b = 1 and a, b ∈ R.
(c) Shade the region of points av + bw where 1 ⩽ a, b ⩽ 2.
5. Find x ∈ R if: √
a) P1 (x, 2, 3), P2 (2, 1, 1) and P1 P2 = 21; b) Q1 (x, x, 1, 5), Q2 (0, 3, 5, 7) and Q1 Q2 = 5.
6. Decide whether the given points are collinear or not:
a) P1 (1, 2, 0, 0), P2 (−2, −2, −3, 0), P3 (7, 10, 6, 0); b) Q1 (2, 3, 2, 0, 1, 0), Q2 (1, 4, 4, 0, 0, 0),
Q3 (5, 0, −4, 0, 4, 4).
7. The coordinates of the midpoint of the segment [P1 P2 ], determined by P1 (x1 , y1 , z1 ) and
P2 (2, 3, 6) are (−1, −4, 8). Find the coordinates of P1 .
8. Let P3 be the midpoint of the segment joining the points P1 (−3, 4, 1) and P2 (−5, 8, 3).
Find the coordinates of the midpoint of the segment: a) joining P1 and P3 ; b) joining P3
and P2 .
9. Compute the area of the triangle whose vertices have coordinates P1 = (−1, 0, 1), P2 =
(0, 2, 2) and P3 = (0, −1, 2).
10. Consider a triangle ABC. Find, with proof, the position vector of its centroid G, with
respect to the position vectors of A, B and C. Suppose the side lengths of the triangle are
a = BC, b = CA and c = AB. Find the position vector of the incenter I of triangle ABC.
11. Is it possible to construct an equilateral triangle ABC in R2 such that the coordinates
of the vertices are all integer numbers?
Date: October 1, 2023.
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