Advertisements
Advertisements
प्रश्न
A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians
Advertisements
उत्तर
`vec"OA" = hat"i", vec"OB" = hat"j", vec"OC" = hat"k"`
D is the midpoit of BC
∴ `vec"OD" = (vec"OB" + "OC")/2`
= `(hat"j" + hat"k")/2`
E is the midpoint of AC
`vec"OE" = (hat"i" + hat"k")/2`
F is the midpoint of AB
∴ `vec"OF" = (hat"i" + hat"j")/2`
Now the medians are `vec"AD", vec"BE"` and `vec"CF"`
(i) `vec"AD" =vec"OD" - vec"OA" = (hat"j" +hat"k")/2 - hat"i"`
= `-hat"i" + hat"j"/2 + hat"k"/2`
`|vec"AD"| = sqrt(1 + 1/4 + 1/4)`
= `sqrt(1 + 1/2)`
= `sqrt(3)/sqrt(2)`
and d.c's of `vec"AD" = ((-1)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2)))`
= `(- sqrt(2)/sqrt(3), sqrt(2)/(2sqrt(3)), sqrt(2)/(2sqrt(3)))`
= `(- sqrt(2)/sqrt(3), 1/(sqrt(2)sqrt(3)), 1/(sqrt(2)/sqrt(3)))`
= `(- sqrt(2)/sqrt(3), 1/sqrt(6), 1/sqrt(6))`
`["Now" sqrt(2)/sqrt(3) = sqrt(2)/sqrt(3) xx sqrt(2)/sqrt(2) = 2/sqrt(6)] = (- 2/sqrt(6), 1/sqrt(6), 1/sqrt(6))`
(ii) `vec"BE" = vec"OE" - vec"OB"`
= `(hat"i" + hat"k")/2 -hat"j"`
= `hat"i"/2 - hat"j" + hat"k"/2`
`|vec"BE"| = sqrt(1/4 + 1 + 1/4)`
= `sqrt(3)/sqrt(2)`
d.c's of `vec"BE" = ((1/2)/(sqrt(3)/sqrt(2)), (-1)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2))) = (1/sqrt(6), (-2)/sqrt(6), 1/sqrt(6))`
(iii) `vec"CE" = vec"OF" - vec"OC"`
= `(hat"i" + hat"j")/2 - hat"k"`
= `hat"i"/2 + hat"j"/2 - hat"k"`
`|vec"CF"| = sqrt(1/4 + 1/4 + 1)`
= `sqrt(3)/sqrt(2)`
d.c's of `vec"CF" = ((1/2)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2)), (-1)/(sqrt(3)/sqrt(2)))`
= `(1/sqrt(6), 1/sqrt(6), (-2)/sqrt(6))`
APPEARS IN
संबंधित प्रश्न
Which of the following represents direction cosines of the line :
(a)`0,1/sqrt2,1/2`
(b)`0,-sqrt3/2,1/sqrt2`
(c)`0,sqrt3/2,1/2`
(d)`1/2,1/2,1/2`
Write the direction ratios of the following line :
`x = −3, (y−4)/3 =( 2 −z)/1`
Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Find the angle between the lines whose direction cosines are given by the equations
2l − m + 2n = 0 and mn + nl + lm = 0
What are the direction cosines of Z-axis?
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is
If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are
Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`, find the magnitude and direction cosines of `vec"a", vec"b", vec"c"`
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is `vec"r" = 3hat"i" + 5hat"j" + 4hat"k" + lambda(2hat"i" + 3hat"j" + 7hat"k")`
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
A line in the 3-dimensional space makes an angle θ `(0 < θ ≤ π/2)` with both the x and y axes. Then the set of all values of θ is the interval ______.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
Equation of line passing through origin and making 30°, 60° and 90° with x, y, z axes respectively, is ______.
If the equation of a line is x = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.
