Arts (English Medium) इयत्ता १२ - CBSE Important Questions for Mathematics

If `hata` and `hatb` are unit vectors, then prove that `|hata + hatb| = 2 cos  theta/2`, where θ is the angle between them.

The scalar projection of the vector `3hati - hatj - 2hatk` on the vector `hati + 2hatj - 3hatk` is ______.

If two vectors `veca` and `vecb` are such that `|veca|` = 2, `|vecb|` = 3 and `veca.vecb` = 4, then `|veca - 2vecb|` is equal to ______.

Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2

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|`.

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.

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`.

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`.

If `|veca`| = 3, `|vecb|` = 5, `|vecc|` = 4 and `veca + vecb + vecc` = `vec0`, then find the value of `(veca.vecb + vecb.vecc + vecc.veca)`.

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`.

If the angle between `veca` and `vecb` is `π/3` and `|veca xx vecb| = 3sqrt(3)`, then the value of `veca.vecb` is ______.

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 ______.

If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.

If `|veca xx vecb| = sqrt(3)` and `veca.vecb` = – 3, then angle between `veca` and `vecb` is ______.

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 ______.

If `veca = hati + hatj + hatk` and `vecb = hati + 2hatj + 3hatk` then find a unit vector perpendicular to both `veca + vecb` and `veca - vecb`.

If `veca.hati = veca.(hati + hatj) = veca.(hati + hatj + hatk)` = 1, then `veca` is ______.

If `veca = 4hati + 6hatj` and `vecb = 3hatj + 4hatk`, then the vector form of the component of `veca` along `vecb` is ______.

Read the following passage and answer the questions given below:

1. What is the magnitude of the force of Team A ?
2. Which team will win the game?
3. Find the magnitude of the resultant force exerted by the teams.
   OR
   In what direction is the ring getting pulled?

Show that the lines `(x+1)/3=(y+3)/5=(z+5)/7 and (x−2)/1=(y−4)/3=(z−6)/5` intersect. Also find their point of intersection