Solve all Q4 explaining detailly each step 1. With respect to the origin 0, the points A, B, C, have position vectors (i+ 2j + k),(3i + 3j + 4k), (-i+ j -k) respectively.a. Show that A. B and C are not collinear. b. Find a vector equation of the plane ABC.c. Find a Cartesian equation for the plane which passes through A and is perpendicular toAB. d. Find the cosine of the angle BAC.2. Vector equations of two lines are given by F (9i+ 4j + 5k)+a (4i -j+ 2k);r = (4i+ 3j + 2k)+ B(-3i+j- k), where a and B are scalar parameters. a) Find, the position vector of the pointsof intersection of the two line. Also find a) a vector equation of the plane which containsboth iines b) a Cartesian equation of the plane which contains both lines.3. A plane contains the points A, B, C whose position vectors, relative to the origin 0, are(2i - j+ k), (3i + 2j – k), (-i + 3j + 2k), respectively. Finda) the Cartesian equation of the plane ABC.b)a vector equation of the line AB.4. Given that the position vectors of the points A, B, C and D relative to the origin O, are: a =3k, b = 2i + 4j + 2k, c= 4i + 3j +k and d= 3i + 7j+ 5k respectively.Finda) a vector equation of the line ABb) Cartesian equation of the line 1, passing through a and d.c) A Cartesian equation of the plane, , containing A, B and Cd) The angle between the plane, n and the line 1 to two decimal places.IT5. i) The angle between two vectors a and b is-Given that /a/= 3 and /b/= 4, find3(a) /2a + 3b/(b) /2a - 3b/

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Answered: a vector equation of the line AB…

a vector equation of the line AB Cartesian equation of the line 1, passing through a and d. A Cartesian equation of the plane, n, containing A, B and C

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter1: Vectors

Section1.3: Lines And Planes

Problem 16EQ

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Question

Solve all Q4 explaining detailly each step

1. With respect to the origin 0, the points A, B, C, have position vectors (i+ 2j + k),
(3i + 3j + 4k), (-i+ j -k) respectively.
a. Show that A. B and C are not collinear. b. Find a vector equation of the plane ABC.
c. Find a Cartesian equation for the plane which passes through A and is perpendicular to
AB. d. Find the cosine of the angle BAC.
2. Vector equations of two lines are given by F (9i+ 4j + 5k)+a (4i -j+ 2k);r = (4i+ 3j + 2k)
+ B(-3i+j- k), where a and B are scalar parameters. a) Find, the position vector of the points
of intersection of the two line. Also find a) a vector equation of the plane which contains
both iines b) a Cartesian equation of the plane which contains both lines.
3. A plane contains the points A, B, C whose position vectors, relative to the origin 0, are
(2i - j+ k), (3i + 2j – k), (-i + 3j + 2k), respectively. Find
a) the Cartesian equation of the plane ABC.
b)a vector equation of the line AB.
4. Given that the position vectors of the points A, B, C and D relative to the origin O, are: a =
3k, b = 2i + 4j + 2k, c= 4i + 3j +k and d= 3i + 7j+ 5k respectively.Find
a) a vector equation of the line AB
b) Cartesian equation of the line 1, passing through a and d.
c) A Cartesian equation of the plane, , containing A, B and C
d) The angle between the plane, n and the line 1 to two decimal places.
IT
5. i) The angle between two vectors a and b is-
Given that /a/= 3 and /b/= 4, find
3
(a) /2a + 3b/
(b) /2a - 3b/

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