chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?


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1 chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " " + DA + AB + BC f)!!! "!!! "!!! "!!! " AB + AD + BA + DC AB + BA + AD + DC AC. Consider the parallelogram shown alongside. Which of the following statements are true? a) AB DC : true b) a b : false c) BC b : true d) AC + CD b : true e) AD CB : false. A triangle ABC is given by AB c and BC a. Point D is the midpoint of the side AB. Express the vectors AC, AD and CD in terms of a and c. " " AC a + c AD c" CD a " + c". In a parallelogram ABCD, let AB a and BC b. Point E is the midpoint of AB ; point F lies on BC so that BF : FC : holds. Express the vectors AE, AC, BD, CD, DE, BF, AF and EF in terms of a and b. AE a" " " AC a + b " " BD b a CD a " DE a" b " BF! "!! b"
2 chapter vector geometry solutions V. " AF a + b" EF a" + b". In a plane quadrilateral ABCD, let AB a, BC b and CD c. a) Express d DA in terms of a, b and c. d! a! + b! + c! ( ) b) Use vectors to show that the midpoints of the sides of the quadrilateral are vertices of a parallelogram. EF a" + b " ( ) HG d" + c " #!!! " CA a" b " Thus: EF HG ( ) ( a" + b " ). Let S be the centre of gravity of a triangle ABC. Prove: SA + SB + SC. SA! MA b" + a" SB c" + b" SC!!! " a" + c" SA + SB + SC!!! " a " + b " + c " " 7. Prove: A quadrilateral whose diagonals bisect each other is a parallelogram. a! + f "! e!! c! f "! + e!! a! + c!! a! c!
3 chapter vector geometry solutions V. Exercise B. Given a! a a. How do you find the magnitude (length) a! of the vector a!? a! a + a (Pythagoras). Given a a a and b b b. How do you calculate a) c a + b? c! a! + b! a a + b b a e! + a e! ( ) + ( b e! + b e! ) ( a + b ) e! + ( a + b ) e! a + b a + b b) c a b? c! a! b! a a b b a e! + a e! ( ) ( b e! + b e! ) ( a b ) e! + ( a b ) e! a b a b c) c k a? c! k a! k a a ( ) k a e! + a e! ( k a ) e! + ( k a ) e! k a k a
4 chapter vector geometry solutions V.!!! ".. Given A( x A / y A ) and B( x B / y B ). Find AB!!! " x b x a AB OA + OB OB OA y b y a (head minus tail). Given a and b. How do you check (without drawing the vectors) if a and b are parallel (or antiparallel) to each other? b! k a! k > a! and b! are parallel k < a! and b! are antiparallel. Let a, b, c. The vectors d a b + c and e x are collinear. Find x. x ( k )
5 chapter vector geometry solutions V.. Complete the parallelogram ABCD where A, B and C are given. What are the coordinates of D? ( ), B( 8 /), C( / ) a) A / OD OA + BC D( /) + b) A( / ), B( 7 / ), C( / 7) D( /8) 7. Express vector c in terms of vectors a and b ( c k a + k b ). a) a c! a! + b! b) a c! a! b! ; b ; b 8 ; c ; c Given A( 9 / ), B( / ), C( /). Calculate the perimeter of the triangle ABC. perimeter: 8 9. Given A( / ), B( 7 / ). Find the coordinates of the midpoint of AB.!!!!!! "! OM AB OA + AB OA + OB OA OA + OB M AB ( /) ( ) ( )
6 chapter vector geometry solutions V.. Given A( / ), B( / 7), C( / ). Find the coordinates of the centre of gravity of the triangle ABC.! OS OA + AB! + M!!!!! " ABC OA + OB ( OA ) + ( OC OM!!!! " AB ) OA + OB ( OA ) + OC OA ( + OB ) OA OA OA + OB OB + OC OA + OB + OC S( / ) ( ) Exercise C. Given a a a a. How do you find the magnitude (length) a of the vector a? a! a a a a + a + a. Determine whether vectors a and b are collinear. a) a a!, b 9 and b! are not collinear, because ( ) and ( ) ( ) but 9
7 chapter vector geometry solutions V. b) a 7, b.. a! and b! are collinear, because b! a!. Determine whether point C lies on the line passing through A and B. a) A( / / ) ; B( / / ) ; C( 8 / 8 / ) yes, because! AC!!! " AB!!! " AB ;! AC b) A( / / ) ; B( / 7 / ) ; C( / / 8) no, because there is no k! so that! AC!!! " k AB!!! " AB ;! AC 9. Determine whether A( / / ), B( / / ) and C ( / 9 / ) are vertices of a triangle ABC. You have to show that A, B and C are on the same line or not.!!! " AB ;! AC 9 9! AC!!! " AB A, B and C are on the same line and therefore A, B and C are not the vertices of a triangle.. The points A( / / ), B( / / ) and C ( / / 7) are vertices of a parallelogram. Determine the coordinates of the fourth vertex D (three results!). D ( / / ); D ( / / ); D / 9 / ( ). Calculate the length of vector a direction and length... Find the components of a vector with the same a! 9; 7. 7
8 chapter vector geometry solutions V. 7. Calculate the length of vector a 9. Find the components of a vector of opposite direction and length of. a! ; Find the points on the z axis where the distance to P( / / 7) is 7. ( ) PQ Q / / z PQ 7 z ;z 9 z 7 Q ( / / );Q ( / / 9) Exercise D. Find the angle between a and b where the following is satisfied: a) a b a b α b) a! b! a! b! α. Calculate the scalar product a b. a) a!, b! a! b! b) a!, b! a! b! 8
9 chapter vector geometry solutions V.. Find the angle between the vectors a and b. a) a, b α 9.9 b) a, b 7 α 8.. Calculate the angle between a and the coordinate axes. a) a α.87;α.8 b) a α 9. ;α.88 ;α 77.. Find the possible values of u given that a and b are perpendicular. a) a 7, b u u b) a u u 7, b u + 9 u + u ;u. Determine the point P given that APB 9. a) A( / / 8), B( / / ), P on the y axis. P ( / / );P ( / / ) b) A( / /), B( / / ), P on the x axis. P ( / / );P ( / / ) 9
10 chapter vector geometry solutions V. 7. The vectors a x and b z form sides of a square. Calculate the area of this square. A 9 with x and z A. with x z. Exercise E. For each of the following determine the vector and the Cartesian equation. The line passes a) through A( / 7) and its gradient m equals to ; vector equation: r! 7 + t Cartesian equation: x y b) through A( /) and B( / ) ; vector equation: r! + t 7 Cartesian equation: x 7y + c) through A( / ) and intersects the y axis at y ; vector equation: r! + t Cartesian equation: x + y d) through A( / 7) and is parallel to the y axis; vector equation: r! 7 + t Cartesian equation: x + e) through A( 8 / ) and is parallel to the x axis. vector equation: r! 8 + t Cartesian equation: y +
11 chapter vector geometry solutions V.. Calculate the point and angle of intersection of the lines l and l. a) l :r + s ; l :r 9 + t S( / 7);ϕ 7.9 b) l :r 8 + s ; l :r + t l l c) l : y x 7 ; l : y 7x S( / );ϕ.9 d) l :x y ; l : x + y S( / );ϕ 9. Show that the lines l :r! + s and l :r! + t are normal. with the dot product of the direction vectors: Exercise F. Find a vector equation for the line that passes through a) A( / / ) and B( / / ) ;! 7 r + t b) A( / / ) and intersects the x axis at x ;! r + t c) A( / / ) and is parallel to the z axis;! r + t
12 chapter vector geometry solutions V. d) A( 7 / / ) and is parallel to the y axis.! 7 r + t. Given the points A( // ) and B( / / ), find a vector equation of the line that passes through the midpoint of the segment AB and is parallel to the x axis. l :r!. + t. Are the points A( / / ) and B( / / 7) on the line l :r A yes, B no + t?. The segment with endpoints A( / / ) and B( / / ) is divided into three equal parts. Find the coordinates of the dividing points. P ( / / ) ; P ( // ). Are the following lines skew, parallel, coincident or intersecting? a) l :r! + s, l :r! + t skew b) l :r.8 + s.., l :r + t parallel c) l :r. + s.., l :r + t identical d) l :r + s, l :r intersecting at S( / / 7) + t
13 chapter vector geometry solutions V. e) l :r 7 + s, l :r + t. parallel. Find the point of intersection S and the acute angle between l :r + s and l :r + s 7. S( / 8 /) ;. 7. In a triangle ABC the vertices A( / / ) and B( 7 / 9 / ) are given. The vertex C lies on the line through P( // ) and Q( // ). Calculate the coordinates of the vertex C given that the side c AB a) is the hypotenuse of the rightangled triangle ABC, C ( // ) ; C ( // 8) b) is the base of the isosceles triangle ABC. C // Exercise G. Determine the Cartesian equation of the plane ABC. a) A( // 7), B( / / ), C( / / ) x + y z b) A( / / ), B( 7 / / ), C ( / / ) x y + z. Determine the Cartesian equation of the following planes given by ( ) and l :r! a) P / / y z t b) A( / /) and B( / / ) ; also, the z axis is parallel to the plane. x 7y
14 chapter vector geometry solutions V.. Determine the Cartesian equation of the plane containing a) l :r! + s and l :r! + t ; x b) P( /. /), parallel to the xz plane; y + c) l :r! 8 + s and l :r! 8 + t. z 8. Describe the particular position of the following planes. a) ε : x 7y + ε z axis b) ε :y z + 9 ε x axis c) ε : x ε yz plane d) ε :x + z ε through y axis. Prove that the lines l and l intersect; determine the Cartesian equation of the plane containing l and l. a) l :r! + s 7, l :r! x y + z ;S( / / ) + t b) l :r! 8 + s, l :r! 8x y z ;S( // ) 8 + t
15 chapter vector geometry solutions V.. Calculate the axes intercepts of the plane ε. a) ε : x y + z a ;b ;c b) ε : x z + a ;c 7. Determine the Cartesian equation for the plane with intercepts x a, y b and z c. Then divide the equation by abc. bcx + acy + abz abc resp. x a + y b + z (axes intercept form for the equations for planes) c 8. A plane is given by its axes intercepts. Determine its Cartesian equation. a) a, b, c x + y + z b) a 8, b, c x y + z c) a, b 7 7x + y + d) a 7 x 7 9. Determine vector equations of the trace lines of plane ε. The trace lines of a plane ε are the lines of intersection between the plane ε and the xy , xz  respectively yz plane. a) ε : x y + z l : x y z + t l : x y z + t l : x y z + t
16 chapter vector geometry solutions V. b) ε : y z + 8 l : x y z + t l : x y z + t l : x y z 8 + t. Find point A in plane ε :r! 9 + s 8 + t a) which lies on the z axis, A( / / ) b) which possesses three equal components, A(. /. /.) c) with horizontal projection A' ( / / ). A( / / ). Determine whether A( //), B( / / ), C( / / ) and D( / / ) are vertices of a quadrilateral. yes ( ε :8x 9y z + 8 ). Find the point at which the line l intersects the plane ε. a) l :r! 8 P( / / ) + t, ε : x y + z b) l :r l ε t, ε : y z + 7
17 chapter vector geometry solutions V. c) l :r + t 7, ε :r! 7 + s + t 8 P( / / ) d) l :r 9 + t, ε :r! 9 + s + t l ε. Determine a vector equation of the line of intersection between planes ε and ε. a) ε : x y + z, ε : x + y z + l : x y z + t b) ε : x + y z +, ε : x y + z l : x y z + t Exercise H. Show in different ways that A( 7 / /), B( 9 / / 7) and C( / 7 / ) lie on the same line.!!! " AB ;! AC 8 I)! AC! AB!!! " AB and AC are collinear II) III)!!! " " AB AC!!! "!!! " AB AC AB AC or!!! "!!! " AB AC AB AC!!! "!!! " here: AB AC AB AC 7
18 chapter vector geometry solutions V.. A line l passes through P / / ( ) and is perpendicular to the lines r + s and r + t. Find the vector equation of l. l :r! + t + t 7 7. Find a vector that is perpendicular to the plane containing the points A( / /), B( / / ) and C ( / / ).!!! " AB ;! AC v! """! """! 9 AB AC or v! 9. Calculate the area of the triangle A( 7 / /), B( / / ), C( 9 / /).. Find the area of the parallelogram having the diagonals e! "! e f and f. Exercise I. Determine the Cartesian equation of the plane that passes through point P and is parallel to the plane ε. a) P( / / ) ; ε :x y + z + x y + z b) P( // ) ; ε : x + z x + z 8
19 chapter vector geometry solutions V.. Determine the Cartesian equation of the plane that contains the point P and is normal to the line l. ( ) ; l :r! a) P / / x + y z + + t ( ) ; l :r b) P / / x + y z 8 + t. Which point on the line l : r + t is equidistant from the points A ( / / ) and B( / / 7)? P( // ). Determine the Cartesian equation of the plane which contains the points P( / / ) and Q( / /) and which is normal to the plane ε : x + y. x y z + 9. Point P( / / ) is reflected in the plane ε : x y + z +. Find the coordinates of the image point P'. P' ( // ). Point P' ( / / 7) is the reflection point of P( / / ). Find the Cartesian equation for the plane ε in which P was reflected. x + y 9z + 7. The line l : r 7 + t is reflected in the plane ε : x + y z +. Determine a vector equation of the reflected line l '. l ' :r! + t 9
20 chapter vector geometry solutions V. 8. A ray of light passes through P( 7 / 7 / ) and is reflected in the plane ε : x y + z. Point Q( 7 / / 8) lies on the reflected ray of light. At which point in the plane ε is the ray of light reflected? R( / / ) 9. Determine the acute angle between the planes ε : x y z + and ε : x + z. α.. Determine the angle of intersection of line l : r + t and plane ε : x y +. α 7. Exercise J. What is the distance between point P( / ) and the line l : x + y? d. Find the length of the perpendicular to the line l : r 9 + t from the origin. d. Calculate the distance between point P( / /) and plane ε : x + 7y z +. d. Given the plane ε : x y 7z +. Find another plane ε passing through point P( / /) and parallel to ε. Determine the Cartesian equation of the plane ε as well as the distance between the two planes. ε : x y 7z, d. Find the distance between the skew lines l and l. a) l :r! + t, l :r! + t d b) l : r + t, l : r + t d
21 chapter vector geometry solutions V.. Determine the Cartesian equation of the plane that is parallel to the plane ε and d units away from it. a) ε :x y +z, d ε :x y +z + and ε :x y +z b) ε : x 7z +, d ε :x 7z + and ε :x 7z 9 7. Determine the Cartesian equation of the angle bisector planes of the planes ε and ε. a) ε : x y z +, ε : x + y z + β :x + y 8z +, β :x y 7 b) ε : x + y + 7z, ε :x y z + 9 β :x y +z + 7, β :9x y z + 8. Which points of the line l : r + t are equidistant from the planes ε : x y + and ε : x + z + 7? P ( / / ), P ( / / ) Exercise K. Determine the centre O and the radius r of the circle c. a) c : x + y 8x + y O( / ) ; r b) c : x + y + x 8y + 8 O / ; r. Determine the points of intersection of the circle c : x + y and the line l : y x +. P ( /) ; P ( 8 / ). Point A( / ) lies on a circle with a radius r. The centre of the circle lies on the line l : y x +. Determine the equation of the circle. c :( x ) + ( y ) ; c :( x +) + ( y )
22 chapter vector geometry solutions V.. Determine the equation of the circle with the centre O( / ) which touches the line l : x y +. ( x ) + ( y ). Find the equation of the circumcircle of the triangle A( / ), B( 8 / ) and C( / ). ( x ) + ( y ). The circle c :( x ) + ( y + ) 9 is touched by circles, each with radius r, whose centres lie on the line l : x 7y +. Find the equations of the circles. c :( x 9) + ( y ) 9 ; c :( x + ) + ( y ) 9 7. The circles c : x + y 9 and c :( x ) + ( y ) '9 have a common chord. What is its length? P (. /.) ; P (. /.) ; P P 8. A circle passes through A( /) and B( / 7) and touches l : x. Find its equation. c :( x ) + ( y + ) 9 ; c :( x + 9) + ( y 8) '89 9. Find the Cartesian equation of the tangent at the point P( / 9) of the circle c :( x ) + ( y + ) 9. x y +. Find the equations for the tangents of the circle c : x + y through the point P( / ). t : x y + ; t : x + y +. Determine the centre O and the radius r of the sphere S. a) S : x + y + z x + y z + O( / / ) ; r b) S : x + y + z + x z + 9 O( / / ) ; r. Find the points of intersection between the sphere S :( x ) + ( y + ) + z 8 and the line l passing through A( 9 / /) and B( / 7 / ). P ( // ) ; P ( / /)
23 chapter vector geometry solutions V.. The spheres S :( x ) + ( y + ) + z and S : x + y + z x y + z + are given. a) Show that S and S touch each other. O O r + r b) Determine the point of contact. P( 7 / / ). Find the equation of the sphere with centre O( / / ) which touches the plane ε : x y + z +. ( x ) + ( y ) + ( z + ). Find the equations of the spheres with centre O( 9 / / ) which touch the sphere S :( x ) + ( y + ) + ( z ) 9. S :( x 9) + ( y ) + ( z ) ; S :( x 9) + ( y ) + ( z ). Point P( / / ) lies on a sphere with radius r. The centre of the sphere lies on the line that passes through A( / / ) and B( / 7 / ). Find the equation of the sphere. S :( x ) + ( y ) + ( z + ) 9 ; S :( x ) + ( y ) + ( z + ) 9 7. Point P( / / ) lies on the sphere with the centre O( / / ) and the radius r. Find the equation of the tangent plane to the sphere at point P. y z 8. Determine the Cartesian equations for the tangent planes to the sphere S :( x ) + ( y ) + ( z + ) 9 which are parallel to the plane ε : x + y z. τ : x + y z 7 ; τ : x + y z + 9. A ray of light, starting at the light source Q( / 8 / 7), travels in the direction of P( / / ) and is reflected in the sphere S :( x ) + ( y + 8) + z. a) Find the point R on the sphere where the reflection takes place. R( / / ) b) Determine a vector equation of the reflected ray of light.! r + t
24 chapter vector geometry solutions V. c) Find the angle between the rays at point R..
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