Advertisements
Advertisements
Question
Show that the points (2, –1, 3), (4, 3, 1) and (3, 1, 2) are collinear
Solution
Let the given points be A(2, –1, 3), B(4, 3, 1) and C(3, 1, 2)
`vec"OA" = 2hat"i" - hat"j" + 3hat"k"`
`vec"OB" = 4hat"i" + 3hat"j" + hat"k"`
`vec"OC" = 3hat"i" + hat"j" + 2hat"k"`
`vec"AB" = vec"OB" - vec"OA"`
= `(4hat"i" + 3hat"j" + hat"k") - (2hat"i" - hat"j" + 3hat"k")`
= `4hat"i" + 3hat"j" + hat"k" - 2hat"i" + hat"j" - 3hat"k"`
`vec"AB" = 2hat"i" + 4hat"j" - 2hat"k"|`
`|vec"AB"| = |2hat"i" + 4hat"j" - 2hat"k"|`
= `sqrt(2^2 + 4^2 + (-2)^2`
= `sqrt(4 +16 + 4)`
= `sqrt(24)`
AB = `sqrt(6 xx 4)`
= `2sqrt(6)`
`vec"BC" = vec"OC" - vec"OB"`
= `(3hat"i" + hat"j" + 2hat"k") - (4hat"i" + 3hat"j" + hat"k")`
= `3hat"i" + hat"j" + 2hat"k" - 4hat"i" - 3hat"j" - hat"k"`
`vec"BC" = -hat"i" - 2hat"j" + hat"k"`
`|vec"BC"| = |-hat"i" - 2hat"j" + hat"k"|`
= `sqrt((-1)^2 + (-2)^2 + 1^2)`
BC = `sqrt(1 + 4 + 1)`
= `sqrt(6)`
`vec"CA" = vec"OC" - vec"OA"`
= `(3hat"i" + hat"j" + 2hat"k") - (2hat"i" + hat"j" + 3hat"k")`
= `3hat"i" + hat"j" + 2hat"k" - 2hat"i" - hat"j" - 3"k"`
`vec"BC" = -hat"i" - 2hat"j" + hat"k"`
`vec"CA" = |hat"i" + 2hat"j" - hat"k"|`
= `sqrt(1^2 + 2^2 + (-1)^2`
CA = `sqrt(1 + 4 + 1)`
= `sqrt(6)`
AB = `2sqrt(6)`, BC = `sqrt(6)`, CA = `sqrt(6)`
BC + CA =`sqrt(6) + sqrt(6) = 2sqrt(6)`
∴ BC + CA = BA = `2sqrt(6)`
Hence the given points A, B, C are collinear.
APPEARS IN
RELATED QUESTIONS
Find `vec"a"*vec"b"` when `vec"a" = 2hat"i" + 2hat"j" - hat"k"` and `vec"b" = 6hat"i" - 3hat"j" + 2hat"k"`
Find the value λ for which the vectors `vec"a"` and `vec"b"` are perpendicular, where `vec"a" = 2hat"i" + 4hat"j" - hat"k"` and `vec"b" = 3hat"i" - 2hat"j" + lambdahat"k"`
If `vec"a", vec"b", vec"c"` are three vectors such that `vec"a" + 2vec"b" + vec"c"` = 0 and `|vec"a"| = 3, |vec"b"| = 4, |vec"c"| = 7`, find the angle between `vec"a"` and `vec"b"`
Show that the vectors `vec"a" = 2hat"i" + 3hat"j" + 3hat"j" + 6hat"k", vec"b" = 6hat"i" + 2hat"j" - 3hat"k"` and `vec"c" = 3hat"i" - 6hat"j" + 6hat"k"` are mutually orthogonal
Let `vec"a", vec"b", vec"c"` be three vectors such that `|vec"a"| = 3, |vec"b"| = 4, |vec"c"| = 5` and each one of them being perpendicular to the sum of the other two, find `|vec"a" + vec"b" + vec"c"|`
Find λ, when the projection of `vec"a" = lambdahat"i" + hat"j" + 4hat"k"` on `vec"b" = 2hat"i" + 6hat"j" + 3hat"k"` is 4 units
Find the magnitude of `vec"a" xx vec"b"` if `vec"a" = 2hat"i" + hat"j" + 3hat"k"` and `vec"b" = 3hat"i" + 5hat"j" - 2hat"k"`
If `vec"a", vec"b", vec"c"` are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is `1/2 |vec"a" xx vec"b" + vec"b" xx vec"c" + vec"c" xx vec"a"|`. Also deduce the condition for collinearity of the points A, B, and C
Let `vec"a", vec"b", vec"c"` be unit vectors such that `vec"a" * vec"b" = vec"a"*vec"c"` = 0 and the angle between `vec"b"` and `vec"c"` is `pi/3`. Prove that `vec"a" = +- 2/sqrt(3) (vec"b" xx vec"c")`
Find the angle between the vectors `2hat"i" + hat"j" - hat"k"` and `hat"i" + 2hat"j" + hat"k"` using vector product
Choose the correct alternative:
The vectors `vec"a" - vec"b", vec"b" - vec"c", vec"c" - vec"a"` are
Choose the correct alternative:
If `lambdahat"i" + 2lambdahat"j" + 2lambdahat"k"` is a unit vector, then the value of `lambda` is
Choose the correct alternative:
If `vec"a"` and `vec"b"` having same magnitude and angle between them is 60° and their scalar product `1/2` is then `|vec"a"|` is
Choose the correct alternative:
Vectors `vec"a"` and `vec"b"` are inclined at an angle θ = 120°. If `vec"a"| = 1, |vec"b"| = 2`, then `[(vec"a" + 3vec"b") xx (3vec"a" - vec"b")]^2` is equal to
Choose the correct alternative:
If `vec"a"` and `vec"b"` are two vectors of magnitude 2 and inclined at an angle 60°, then the angle between `vec"a"` and `vec"a" + vec"b"` is
Choose the correct alternative:
If the projection of `5hat"i" - hat"j" - 3hat"k"` on the vector `hat"i" + 3hat"j" + lambdahat"k"` is same as the projection of `hat"i" + 3hat"j" + lambdahat"k"` on `5hat"i" - hat"j" - 3hat"k"`, then λ is equal to
Choose the correct alternative:
If `vec"a" = hat"i" + 2hat"j" + 2hat"k", |vec"b"|` = 5 and the angle between `vec"a"` and `vec"b"` is `pi/6`, then the area of the triangle formed by these two vectors as two sides, is
