Advertisements
Advertisements
प्रश्न
Show that the vectors `-hat"i" - 2hat"j" - 6hat"k", 2hat"i" - hat"j" + hat"k"` and find `-hat"i" + 3hat"j" + 5hat"k"` form a right angled triangle
उत्तर
Let `vec"AB"| = -hat"i" - 2hat"j" - 6hat"k"`
`vec"BC" = 2hat"i" - hat"j" + hat"k"`
and `vec"CA" = -hat"i" - 2hat"j" - 5hat"k"`
`|vec"AB"| = |-hat"i" - 2hat"j" - 6hat"k"|`
= `sqrt((-1)^2 + (-2)^2 + (-6)^2`
AB = `sqrt(1 + 4 + 36)`
= `sqrt(41)`
`|vec"BC"| = |2hat"i" - hat"j" + hat"k"|`
= `sqrt(2^2 + (-1)^2 + 1^2)`
BC = `sqrt(4 + 1 + 1)`
= `sqrt(6)`
`|vec"CA"| = |-hat"i" + 3hat"j" + 5hat"k"|`
= `sqrt((-1)^2 + 3^2 + 5^2)`
CA = `sqrt(1 + 9 + 25)`
= `sqrt(35)`
AB ≠ BC + CA
∴ The given vectors form a triangle, Also
AB2 = 41, BC2 = 6, CA2 = 35
AB2 = BC2 + CA2
∴ ∆ABC is a right angled triangle.
APPEARS IN
संबंधित प्रश्न
If `|vec"a"|= 5, |vec"b"| = 6, |vec"c"| = 7` and `vec"a" + vec"b" + vec"c" = vec"0"`, find `vec"a" * vec"b" + vec"b" *vec"c" + vec"c" * vec"a"`
If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that
`sin theta/2 = 1/2|vec"a" - vec"b"|`
If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that
`tan theta/2 = |vec"a" - vec"b"|/|vec"a" + vec"b"|`
Find the angle between the vectors
`hat"i" - hat"j"` and `hat"j" - hat"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"`
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 the projection of the vector `hat"i" + 3hat"j" + 7hat"k"` on the vector `2hat"i" + 6hat"j" + 3hat"k"`
Find the area of the parallelogram whose two adjacent sides are determined by the vectors `hat"i" + 2hat"j" + 3hat"k"` and `3hat"i" - 2hat"j" + hat"k"`
For any vector `vec"a"` prove that `|vec"a" xx hat"i"|^2 + |vec"a" xx hat"j"|^2 + |vec"a" xx hat"k"|^2 = 2|vec"a"|^2`
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:
A vector makes equal angle with the positive direction of the coordinate axes. Then each angle is equal to
Choose the correct alternative:
If `|vec"a" + vec"b"| = 60, |vec"a" - vec"b"| = 40` and `|vec"b"| = 46`, then `|vec"a"|` is
Choose the correct alternative:
The value of θ ∈ `(0, pi/2)` for which the vectors `"a" = (sin theta)hat"i" = (cos theta)hat"j"` and `vec"b" = hat"i" - sqrt(3)hat"j" + 2hat"k"` are perpendicular, equaal to
Choose the correct alternative:
If `|vec"a"| = 13, |vec"b"| = 5` and `vec"a" * vec"b"` = 60° then `|vec"a" xx vec"b"|` 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 `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
