Advertisements
Advertisements
प्रश्न
Find the angle between the vectors `2hat"i" + hat"j" - hat"k"` and `hat"i" + 2hat"j" + hat"k"` using vector product
उत्तर
Let the given vector be `2hat"i" + hat"j" - hat"k"` and `hat"i" + 2hat"j" + hat"k"`
`vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(2, 1, -1),(1, 2, 1)|`
= `hat"i"(1 + 2) - hat"j"(2 + 1) + hat"k"(4 - 1)`
`vec"a" xx vec"b" = 3hat"i" - 3hat"j" + 3hat"k"`
`|vec"a" xx vec"b"| = |3hat"i" - 3hat"j" + 3hat"k"|`
= `sqrt(3^2 + (- 3)^2 + 3^2`
= `sqrt(3 xx 3^2)`
= `3sqrt(3)`
`|vec"a"| = |2hat"i" + hat"j" - hat"k"|`
= `sqrt(2^2 + 1^2 + (- 1)^2`
= `sqrt(4 + 1 + 1)`
= `sqrt(6)`
`|vec"b"| = |hat"i" + 2hat"j" - hat"k"|`
= `sqrt(1^2 + 2^2 + 1^2)`
= `sqrt(1 + 4 + 1)`
= `sqrt(6)`
Let θ be the angle between `vec"a"` and `vec"b"`
sin θ = `|vec"a" xx vec"b"|/(|vec"a"| |vec"b"|)`
= `(3sqrt(3))/(sqrt(6) * sqrt(6))`
= `(3sqrt(3))/6`
sin θ = `sqrt(3)/2`
∴ θ = `pi/3`
APPEARS IN
संबंधित प्रश्न
Show that the vectors `- 2hat"i" - hat"j" - hat"k", - 3hat"i" - 4hat"j" - 4hat"k", hat"i" - 3hat"j" - 5hat"k"` form a right angled triangle
Show that the points (2, –1, 3), (4, 3, 1) and (3, 1, 2) are collinear
If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that
`cos theta/2 = 1/2|vec"a" + vec"b"|`
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 `-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
Three vectors `vec"a", vec"b"` and `vec"c"` are such that `|vec"a"| = 2, |vec"b"| = 3, |vec"c"| = 4`, and `vec"a" + vec"b" + vec"c" = vec0`. Find `4vec"a"*vec"b" + 3vec"b"*vec"c" + 3vec"c"*vec"a"`
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"`
Show that `vec"a" xx (vec"b" + vec"c") + vec"b" xx (vec"c" + vec"a") + vec"c" xx (vec"a" + vec"b") = vec0`
Find the vectors of magnitude `10sqrt(3)` that are perpendicular to the plane which contains `hat"i" + 2hat"j" + hat"k"` and `hat"i" + 3hat"j" + 4hat"k"`
Find the unit vectors perpendicular to each of the vectors `vec"a" + vec"b"` and `vec"a" - vec"b"`, where `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"b" = hat"i" + 2hat"j" + 3hat"k"`
Find the area of the triangle whose vertices are A(3, – 1, 2), B(1, – 1, – 3) and C(4, – 3, 1)
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")`
Choose the correct alternative:
A vector `vec"OP"` makes 60° and 45° with the positive direction of the x and y axes respectively. Then the angle between `vec"OP"` and the z-axis is
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:
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:
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
