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The value of λ for which the two vectors `2hati - hatj + 2hatk` and `3hati + λhatj + hatk` are perpendicular is ______.
Concept: Product of Two Vectors >> Vector (Or Cross) Product of Two Vectors
If `hata` and `hatb` are unit vectors, then prove that `|hata + hatb| = 2 cos theta/2`, where θ is the angle between them.
Concept: Product of Two Vectors >> Scalar (Or Dot) Product of Two Vectors
Find the direction cosines of the following line:
`(3 - x)/(-1) = (2y - 1)/2 = z/4`
Concept: Direction Cosines
The scalar projection of the vector `3hati - hatj - 2hatk` on the vector `hati + 2hatj - 3hatk` is ______.
Concept: Product of Two Vectors >> Projection of a Vector on a Line
If two vectors `veca` and `vecb` are such that `|veca|` = 2, `|vecb|` = 3 and `veca.vecb` = 4, then `|veca - 2vecb|` is equal to ______.
Concept: Product of Two Vectors >> Scalar (Or Dot) Product of Two Vectors
Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2
Concept: Basic Concepts of Vector Algebra
Let `veca = hati + hatj, vecb = hati - hatj` and `vecc = hati + hatj + hatk`. If `hatn` is a unit vector such that `veca.hatn` = 0 and `vecb.hatn` = 0, then find `|vecc.hatn|`.
Concept: Product of Two Vectors >> Vector (Or Cross) Product of Two Vectors
If `veca` and `vecb` are unit vectors inclined at an angle 30° to each other, then find the area of the parallelogram with `(veca + 3vecb)` and `(3veca + vecb)` as adjacent sides.
Concept: Product of Two Vectors >> Vector (Or Cross) Product of Two Vectors
Write the projection of the vector `(vecb + vecc)` on the vector `veca`, where `veca = 2hati - 2hatj + hatk, vecb = hati + 2hatj - 2hatk` and `vecc = 2hati - hatj + 4hatk`.
Concept: Product of Two Vectors >> Projection of a Vector on a Line
If `veca, vecb, vecc` are three vectors such that `veca.vecb = veca.vecc` and `veca xx vecb = veca xx vecc, veca ≠ 0`, then show that `vecb = vecc`.
Concept: Properties of Vector Addition
If `|veca`| = 3, `|vecb|` = 5, `|vecc|` = 4 and `veca + vecb + vecc` = `vec0`, then find the value of `(veca.vecb + vecb.vecc + vecc.veca)`.
Concept: Properties of Vector Addition
If `veca, vecb, vecc` are three non-zero unequal vectors such that `veca.vecb = veca.vecc`, then find the angle between `veca` and `vecb - vecc`.
Concept: Product of Two Vectors >> Scalar (Or Dot) Product of Two Vectors
If the angle between `veca` and `vecb` is `π/3` and `|veca xx vecb| = 3sqrt(3)`, then the value of `veca.vecb` is ______.
Concept: Product of Two Vectors >> Vector (Or Cross) Product of Two Vectors
The position vectors of three consecutive vertices of a parallelogram ABCD are `A(4hati + 2hatj - 6hatk), B(5hati - 3hatj + hatk)`, and `C(12hati + 4hatj + 5hatk)`. The position vector of D is given by ______.
Concept: Section Formula
If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.
Concept: Basic Concepts of Vector Algebra
If `|veca xx vecb| = sqrt(3)` and `veca.vecb` = – 3, then angle between `veca` and `vecb` is ______.
Concept: Product of Two Vectors >> Vector (Or Cross) Product of Two Vectors
A unit vector `hata` makes equal but acute angles on the coordinate axes. The projection of the vector `hata` on the vector `vecb = 5hati + 7hatj - hatk` is ______.
Concept: Product of Two Vectors >> Projection of a Vector on a Line
If `veca = hati + hatj + hatk` and `vecb = hati + 2hatj + 3hatk` then find a unit vector perpendicular to both `veca + vecb` and `veca - vecb`.
Concept: Product of Two Vectors >> Vector (Or Cross) Product of Two Vectors
If `veca.hati = veca.(hati + hatj) = veca.(hati + hatj + hatk)` = 1, then `veca` is ______.
Concept: Product of Two Vectors >> Scalar (Or Dot) Product of Two Vectors
If `veca = 4hati + 6hatj` and `vecb = 3hatj + 4hatk`, then the vector form of the component of `veca` along `vecb` is ______.
Concept: Components of Vector
