Advertisements
Advertisements
Question
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
Solution
`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
RELATED QUESTIONS
Find the direction cosines of a line which makes equal angles with the coordinate axes.
If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1, n1l2 − n2l1, l1m2 − l2m1.
Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).
What are the direction cosines of Z-axis?
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
Answer each of the following questions in one word or one sentence or as per exact requirement of the question:
Write the distance of a point P(a, b, c) from x-axis.
A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is
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
The distance of the point P (a, b, c) from the x-axis is
Verify whether the following ratios are direction cosines of some vector or not
`1/sqrt(2), 1/2, 1/2`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
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 `3vec"a"- 2vec"b"+ 5vec"c"`
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
A line passes through the points (6, –7, –1) and (2, –3, 1). The direction cosines of the line so directed that the angle made by it with positive direction of x-axis is acute, are ______.
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.
Find the coordinates of the foot of the perpendicular drawn from point (5, 7, 3) to the line `(x - 15)/3 = (y - 29)/8 = (z - 5)/-5`.
