Find the direction cosines of the sides of the triangle whose vertices are (3, 5, –4), (–1, 1, 2) and (–5, –5, –2).
Let A(3, 5, – 4), B(–1, 1, 2), C(–5, –5, –2) be the vertices of ΔABC. Direction ratios of AB are – 1 – 3, 1 – 5, 2 + 4 i.e. – 4, – 4, 6 cosines of the line AB as i.e. Direction ratios of BC are – 5 + 1, –5 –1, –2 –2 i.e. – 4, –6, –4. Dividing each by direction ratios of the line BC as Direction ratios of CA are 3+5, 5+5, -4+2 i.e., 8, 10 -2. Dividing each by direction ratios of the line CA as
Direction Ratios of a Line Any 3 numbers which are proportional to the direction cosines of a line are known as direction ratios of that line. Direction ratios are denoted by (a,b and c). Consider a vector OP-->=a î +bĵ +ck̂ (Where a, b and c are direction ratios. ) =r (cos α î + cosβ ĵ + cos γ k̂) where r=√ (a2+b2+c2) cosα,cosβ and cosγ are direction cosines. r cosα=a,r cosβ=b, r cosγ=c are direction ratios. Relation between Direction Ratios and Direction Cosines Consider a line whose one point is at originand another is at P. Considering a vector OP-->=a î +bĵ +ck̂ (equation(1)) =r cos αî+r cosβĵ+r cos γk̂ where r=√ (a2+b2+c2) (equation(2)) Comparing equations (1) and (2) :- a=(rcos α) ,b =(r cosβ) and c=(rcosγ)(equation(3)) cos α =(a/r), cosβ=(b/r) and cosγ=(c/r) (equation(4)) Squaring and adding equation(4) :- (a2/r2) + (b2/r2)+(c2/r2) =1 using (cos2 α + cos2 β + cos2 γ) =1 =>(a2+ b2+c2) = r2 (equation(5)) From equation(4) cos α =(a/r)= (±) (a)/(√( a2+ b2+c2) using equation(5) Similarly cos β=(b/r) =(±)(b)/ (√( a2+ b2+c2) and cos γ=(c/r) =(±)(c)/ (√( a2+ b2+c2) usingequation (5) Therefore cos α, cos β and cosγare the direction cosines which are written in the form of a, b and c which are direction ratios.
Relationship between direction ratios and cosines when a line passes through 2 points One and only one line passes through two given points. Consider a line PQ and the coordinates of points P and Q are given as P(x1, y1, z1) and Q(x2,y2,z2). Let l,m and n are the direction cosines of the line PQ and which makes angleα,βand γ with the x,y and z-axis respectively. If we draw perpendiculars from the point P and Q as shown in the figure (b), to XY-plane to meet at R and S. Also drawing a perpendicular from P to QS to meet at N. Therefore cosα= (x2-x1)/(PQ) Similarly cosβ =(y2-y1)/(PQ) and cosγ =(NQ)/(PQ) =(z2-z1)/(PQ) Therefore the direction cosines of the line segment joining the points P(x1, y1, z1) and Q(x2, y2, z2) will be:- PQ=(x2-x1)/(PQ) ,(y2-y1)/(PQ) +(z2-z1)/(PQ) equation(1) Where PQ=√( x2-x1)2+( y2-y1)2+( z2-z1)2 Note:- The direction ratios of the line segment joining point P(x1, y1, z1) and Q(x2, y2, z2) can be considered as – (x2 – x1, y2 – y1, z2 – z1 or x1 – x2, y1 – y2, z1 – z2)
Problem:- If a line makes an angle of 90°, 135o, 45o with the positive direction ofx, y, z-axes, respectively, then find its direction cosines. Answer:- Let l, m and be direction cosines of the line. Then l= cos90° = 0, m=cos 135o =-(1/√2) and n=cos 45o= (1/√2) Therefore Direction cosines: [0,-(1/√2),(1/√2)] Problem:- Find the direction cosines of a line which makes equal angles with the coordinate axes. Answer:- Let the direction cosines of the line make an angle α with each of the coordinate axes. l = cos α, m = cos α, n = cos α l2+m2+n2 =1 =>cos2 α + cos2 β + cos2 γ) =1 =>3cos2 α =1 =>cos2 α = (1/3) =>cosα = (±) (1/√3) Thus, the direction cosines of the line, which is equally inclined to the coordinate axes,are(±)(1/√3),(±) (1/√3),and (±) (1/√3). Problem:- If a line has the direction ratios −18, 12, −4, then what are its direction cosines? Answer:- If a line has direction ratios of −18, 12, and −4, then its direction cosines are = (-18)/ (√ ((18)2+ (12)2+ (-4)2), = (12)/ ((√ ((18)2+ (12)2+ (-4)2), = (-4)/(√ ((18)2+ (12)2+ (-4)2) = i.e. (-18)/ (22), (12)/ (22) and (-4)/ (22) = (-9/11), (6/11) and (-2)/ (11) Thus, the direction cosines are (-9/11), (6/11) and (-2)/ (11). Problem:- Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear. Answer:- The given points are A (2, 3, 4), B (− 1, − 2, 1), and C (5, 8, 7). It is known that the direction ratios of line joining the points, (x1, y1, z1) and (x2, y2, z2),are given by, (x2 − x1), (y2 − y1), and (z2 − z1). The direction ratios of AB are (−1 − 2), (−2 − 3), and (1 − 4) i.e., (−3, −5, and −3). The direction ratios of BC are (5 − (− 1)), (8 − (− 2)), and (7 − 1) i.e., (6, 10, and 6). It can be seen that the direction ratios of BC are −2 times that of AB i.e., they areproportional. Therefore, AB is parallel to BC. Since point B is common to AB and BC, points A, B, and C are collinear. Problem:- Find the direction cosines of the sides of the triangle whose vertices are (3, 5, − 4), (−1, 1, 2) and (− 5, − 5, − 2)? Answer:- The vertices of ∆ABC are A (3, 5, −4), B (−1, 1, 2), and C (−5, −5, −2). The direction ratios of side AB are (−1 − 3), (1 − 5), and (2 − (−4)) i.e., (−4), (−4), and(6). Then,√ ((-4)2+ (-4)2+ (6)2) =√ (16+16+32) =√ (68) =2√ (17) Therefore, the direction cosines of AB are = (-4)/ (√ (-4)2 + (-4)2 + (6)2), = (-4)/ (√ (-4)2 + (-4)2 + (6)2), = (6)/ ((√ (-4)2 + (-4)2 + (6)2) = (-4)/ (2√ (17), - (4)/2√ (17), (-6)/2√ (17) = (-2)/√ (17), (-2)/√ (17), (3)/√ (17) The direction ratios of BC are (−5 − (−1)), (−5 − 1), and (−2 − 2) i.e., (−4), (−6), and (−4). Therefore, the direction cosines of BC are = (-4)/ (√ (-4)2 + (-4)2 + (6)2), (-4)/ (√ (-4)2 + (-4)2 + (6)2), (6)/ ((√ (-4)2 + (-4)2 + (6)2) = (-4)/ (2√ (17), (-4)/(2√ (17)), (-6)/(2√ (17)) The direction ratios of CA are (−5 − 3), (−5 − 5), and (−2 − (−4)) i.e., (−8), (−10), and (2). Therefore, the direction cosines of AC are = (-8)/√ ((-8)2+ (10)2+ (2)2), = (-5)/ ((-8)2+ (10)2+ (2)2), = (2)/ ((-8)2+ (10)2+ (2)2) =(-8)/ (2√ (42)), (-10)/ (2√ (42)), (2)/2√42)) |