(1) Find the direction vector of the line joining the points (a) A(7,6) and B(-2,5), (b)...

 

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(1) Find the direction vector of the line joining the points (a) A(7,6) and B(-2,5), (b) with position vectors -3i- 4j and -2i + 7j, (с) Р(4,-2) аnd Q(4,9). (2) Find the vector equation of the line (a) through the point (1, -5) and parallel to the r-axis; (b) through (2, 1) and (-3, 8); (c) through the point (2,-8) and with gradient -4/5; (d) through (6, -3) and parallel to the line through (5,-1) and (2, 11); (e) through (1, -1) and parallel to the median from A of the triangle A(2,3), B(6, 1), C(7,3). (3) The position vectors of the points A and B are given by a = 2i+3j and b= 4i+j respectively. Write down a vector equation of the line AB and find the value of a if the point C with position vector e = 7i + aj lies on AB (4) Write down the position vectors of any two points on the line r (3i - 4j) + 2 = 0 and hence find the vector equation of the line. (5) Find the vector equation of the line that passes through the point (1, 5) and is parallel to the line r (3+7t)i + (1- 3t)j. (6) Show that the vector equations %3D r = -2i +j+t(-i+5j) and r= -3i + 6j + s(-3i + 15j) refer to the equation of the same line and find the relation between s and t to give the position vector of the same point. (7) Find the angles between the following lines (a) r =i- 2j + (i – j) and r= 2ị – 6j + µ(i – 2j); (b) r i+ 2j + s(2i + j) and r = 4ị + 2j + t(7i – 2j). (8) If the direction vector of a line is 3i + 2j and the gradient of another line is -5/2, find the acute angle between the two lines. (9) Express the equations of the lines r (i- j) +7 = 0, r (i+ 3j) - 5 0 in parametric forms and hence find the position vector of their point of intersection. (1) Find the direction vector of the line joining the points (a) A(7,6) and B(-2,5), (b) with position vectors -3i- 4j and -2i + 7j, (с) Р(4,-2) аnd Q(4,9). (2) Find the vector equation of the line (a) through the point (1, -5) and parallel to the r-axis; (b) through (2, 1) and (-3, 8); (c) through the point (2,-8) and with gradient -4/5; (d) through (6, -3) and parallel to the line through (5,-1) and (2, 11); (e) through (1, -1) and parallel to the median from A of the triangle A(2,3), B(6, 1), C(7,3). (3) The position vectors of the points A and B are given by a = 2i+3j and b= 4i+j respectively. Write down a vector equation of the line AB and find the value of a if the point C with position vector e = 7i + aj lies on AB (4) Write down the position vectors of any two points on the line r (3i - 4j) + 2 = 0 and hence find the vector equation of the line. (5) Find the vector equation of the line that passes through the point (1, 5) and is parallel to the line r (3+7t)i + (1- 3t)j. (6) Show that the vector equations %3D r = -2i +j+t(-i+5j) and r= -3i + 6j + s(-3i + 15j) refer to the equation of the same line and find the relation between s and t to give the position vector of the same point. (7) Find the angles between the following lines (a) r =i- 2j + (i – j) and r= 2ị – 6j + µ(i – 2j); (b) r i+ 2j + s(2i + j) and r = 4ị + 2j + t(7i – 2j). (8) If the direction vector of a line is 3i + 2j and the gradient of another line is -5/2, find the acute angle between the two lines. (9) Express the equations of the lines r (i- j) +7 = 0, r (i+ 3j) - 5 0 in parametric forms and hence find the position vector of their point of intersection.