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(1) Find the normal vectors of the lines joining the points with the following position vectors (a) 4ị + 7j, 6ị – 3j (b) –2į – 3j, -8ị – 7j (c) 2i + j/2, 6+į/6

(1) Find the normal vectors of the lines joining the points with the following position vectors (a) 4ị + 7j, 6ị – 3j (b) –2į – 3j, -8ị – 7j (c) 2i + j/2, 6+į/6

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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The lines r - n1 + ci = 0, r · n2 + cz = 0 meet at the point A. Find the equation of the line through A which is at right angles to the line r · n1 +¢i = 0. (10) If ABC is a triangle whose position vectors of A, B and C are 3i + j, -i + 7j, i – 3j respectively, find, in normal form, the equations of the medians. (11) The equations of the sides of the sides AB, AD of a parallelo- gram ABCD are r: (2i + j) + 5 = 0, r · (i – 3j) + 7 = 0, and C is the point (1,2). Find the equations of (a) the sides BC, CD; (b) the diagonals AC; BD (12) ÅBCD is a parallelogram, the position vectors of A, B and C being 4i + 8j, 3i + 6j and –5į – 2j respectively. Find (a) the normal vectors of AB and BC; (b) the cartesian equations of AD and CD; (c) the position vector of D; (d) the position vector of the point of intersection of AC and BD; (13) ABCD is a parallelogram. The equations of AB, BC are r: (3i – j) + 7 = 0, r · (3i – 3j) + 5 = 0. Find the position vector of the point D, given that AD passes through the point whose position vector is -8i + 3j and CD passes through the point with position vector 5i + 23. (14) The point A(1,2) is one vertex of a parallelogram ABCD where side AB lies on the line r - (i – 2j)+3= 0 and the diagonal BD lies on the line r - (į+3j) – 17 = 0. Given that |AD| = |BĎ|, find the position vector of C and the equations of the sides of the parallelogram through C. (15) Show that the lines r: (4i + 3) – 1 = 0, r ·(-2i + j) – 1 = 0, r · (i + 2j) – 1 = 0 are concurrent. (16) If the lines -2r + 3y = 6, 3r + 5y = 10, 3r – 7y =t are concurrent, find the value of t. (17) Find the equations of the two lines through the poin which make an angle with the line 2r – y = 2.
Transcribed Image Text:The lines r - n1 + ci = 0, r · n2 + cz = 0 meet at the point A. Find the equation of the line through A which is at right angles to the line r · n1 +¢i = 0. (10) If ABC is a triangle whose position vectors of A, B and C are 3i + j, -i + 7j, i – 3j respectively, find, in normal form, the equations of the medians. (11) The equations of the sides of the sides AB, AD of a parallelo- gram ABCD are r: (2i + j) + 5 = 0, r · (i – 3j) + 7 = 0, and C is the point (1,2). Find the equations of (a) the sides BC, CD; (b) the diagonals AC; BD (12) ÅBCD is a parallelogram, the position vectors of A, B and C being 4i + 8j, 3i + 6j and –5į – 2j respectively. Find (a) the normal vectors of AB and BC; (b) the cartesian equations of AD and CD; (c) the position vector of D; (d) the position vector of the point of intersection of AC and BD; (13) ABCD is a parallelogram. The equations of AB, BC are r: (3i – j) + 7 = 0, r · (3i – 3j) + 5 = 0. Find the position vector of the point D, given that AD passes through the point whose position vector is -8i + 3j and CD passes through the point with position vector 5i + 23. (14) The point A(1,2) is one vertex of a parallelogram ABCD where side AB lies on the line r - (i – 2j)+3= 0 and the diagonal BD lies on the line r - (į+3j) – 17 = 0. Given that |AD| = |BĎ|, find the position vector of C and the equations of the sides of the parallelogram through C. (15) Show that the lines r: (4i + 3) – 1 = 0, r ·(-2i + j) – 1 = 0, r · (i + 2j) – 1 = 0 are concurrent. (16) If the lines -2r + 3y = 6, 3r + 5y = 10, 3r – 7y =t are concurrent, find the value of t. (17) Find the equations of the two lines through the poin which make an angle with the line 2r – y = 2.
(1) Find the normal vectors of the lines joining the points with the following position vectors (a) 4ị + 7j, 6ị – 3j (b) –2į – 3j, -8į – 7j (c) 2i + j/2, 6+ į/6 (2) Write down in normal form the equation of the line which is (a) parallel to the line r · (5i +3j) = 2 and passes through the point (2, –3); (b) perpendicular to the line 7x + 6y = 1 and passes through the point (1, 1); (c) passes through the points (3, 1) and (–2, –7); (3) Find the equation of the perpendicular bisector of the line join- ing the points with position vectors (a) 3i + 4j, 7i + 1oj (b) –4į – 3j, 2i – 6j. (4) In each of the following cases write down the normal vectors of the given lines and hence find the acute angles between them: (a) 2r – 5y – 3 = 0, 3x+ 7y = 10; (b) 2r + 3y – 2 = 0, 2r – 8y + 19 = 0, (c) 4y – 7y + 13 = 0, 2x – 8y + 19 = 0 (5) Use vector methods to find the position vectors of the points of intersection of the following pairs of lines: (a) r (i – j) +2 = 0, r·(3i+2j)+6 = 0; (b) r· (4i – j) – 3 = 0, r-(i+2j) = 3; (c) 3r – y+4= 0, 3x – 4y – 15 = 0, (d) 7r – 4y +1= 0, 1- y+1=0. (6) Find the angles of the triangle whose sides are the lines r+2y+ 5 = 0, r – 3y + 4 = 0 and 3r + y +7=0. (7) Find the position vector of the centroid of the triangle formed by the lines r-(i- j) – 2 = 0, r · (3i + j) – 10 = 0, r · (7i – 3j) – 2 = 0. (8) Find the equation of the line perpendicular to the line 4x – 3y = 0 and passes through the point of intersection of the lines I+ 2y – 5 = 0, 3r – y – 1 = 0.
Transcribed Image Text:(1) Find the normal vectors of the lines joining the points with the following position vectors (a) 4ị + 7j, 6ị – 3j (b) –2į – 3j, -8į – 7j (c) 2i + j/2, 6+ į/6 (2) Write down in normal form the equation of the line which is (a) parallel to the line r · (5i +3j) = 2 and passes through the point (2, –3); (b) perpendicular to the line 7x + 6y = 1 and passes through the point (1, 1); (c) passes through the points (3, 1) and (–2, –7); (3) Find the equation of the perpendicular bisector of the line join- ing the points with position vectors (a) 3i + 4j, 7i + 1oj (b) –4į – 3j, 2i – 6j. (4) In each of the following cases write down the normal vectors of the given lines and hence find the acute angles between them: (a) 2r – 5y – 3 = 0, 3x+ 7y = 10; (b) 2r + 3y – 2 = 0, 2r – 8y + 19 = 0, (c) 4y – 7y + 13 = 0, 2x – 8y + 19 = 0 (5) Use vector methods to find the position vectors of the points of intersection of the following pairs of lines: (a) r (i – j) +2 = 0, r·(3i+2j)+6 = 0; (b) r· (4i – j) – 3 = 0, r-(i+2j) = 3; (c) 3r – y+4= 0, 3x – 4y – 15 = 0, (d) 7r – 4y +1= 0, 1- y+1=0. (6) Find the angles of the triangle whose sides are the lines r+2y+ 5 = 0, r – 3y + 4 = 0 and 3r + y +7=0. (7) Find the position vector of the centroid of the triangle formed by the lines r-(i- j) – 2 = 0, r · (3i + j) – 10 = 0, r · (7i – 3j) – 2 = 0. (8) Find the equation of the line perpendicular to the line 4x – 3y = 0 and passes through the point of intersection of the lines I+ 2y – 5 = 0, 3r – y – 1 = 0.
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