Advertisements
Advertisements
प्रश्न
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"`
Advertisements
उत्तर
`3vec"a"- 2vec"b"+ 5vec"c" = 3(2hat"i" + hat"j" - 4hat"k") -2(3hat"i" - 4hat"j" - 5hat"k") + 5(-3hat"i" + 2hat"j" + 3hat"k")`
= `6hat"i" + 9hat"j" - 12hat"k" - 6hat"i" + 8hat"j" + 10hat"k" - 15hat"i" + 10hat"j" + 15hat"k"`
`3vec"a"- 2vec"b"+ 5vec"c" = -15hat"i" + 27hat"j" + 13hatk"`
`|3vec"a"- 2vec"b"+ 5vec"c"| = |-15hat"i" + 27hat"j" + 13hatk"|`
= `sqrt((-1)^2 + (27)^2 + 13^2`
=`sqrt(225 + 729 + 169)`
`|3vec"a"- 2vec"b"+ 5vec"c"| = sqrt(1123)`
Direction cosines of the vector `3vec"a"- 2vec"b"+ 5vec"c"` are
`[(-15)/|-15hat"i" + 27hat"j" + 1hat"k"|, 27/|-15hat"i" + 27hat"j" + 13hat"k"|, 13/|-15hat"i" + 27hat"j" + 13hat"k"|`
`[(-15)/sqrt(113), 27/sqrt(1123), 13/sqrt(123)]`
∴ The magnitude and direction cosines of the vector `3vec"a"- 2vec"b"+ 5vec"c"` are
`sqrt(1123), [(-15)/sqrt(113), 27/sqrt(1123), 13/sqrt(123)]`
APPEARS IN
संबंधित प्रश्न
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
If l, m, n are the direction cosines of a line, then prove that l2 + m2 + n2 = 1. Hence find the
direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.
If a line makes angles 90°, 135°, 45° with the X, Y, and Z axes respectively, then its direction cosines are _______.
(A) `0,1/sqrt2,-1/sqrt2`
(B) `0,-1/sqrt2,-1/sqrt2`
(C) `1,1/sqrt2,1/sqrt2`
(D) `0,-1/sqrt2,1/sqrt2`
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.
Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
If a unit vector `vec a` makes an angle \[\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}\] and an acute angle θ with \[\hat{ k} \] ,then find the value of θ.
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line `vec("r") = (-2hat"i"+3hat"j") +lambda(2hat"i"-3hat"j"+6hat"k").`Also, find the distance between these two lines.
Find the direction cosines and direction ratios for the following vector
`hat"j"`
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")`
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
Find the direction cosine of a line which makes equal angle with coordinate axes.
If two straight lines whose direction cosines are given by the relations l + m – n = 0, 3l2 + m2 + cnl = 0 are parallel, then the positive value of c is ______.
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`.
If a line makes an angle α, β and γ with positive direction of the coordinate axes, then the value of sin2α + sin2β + sin2γ will be ______.
