Question

Two honeybees are flying parallel to each other in the garden to collect the nectar. The path traced by the bees is given in the form of a straight line. The equation of the path traced by one honeybee is `vecr = (hati + 2hatj + 3hatk) + λ(2hati + 3hatj + 4hatk)`.

1. Write the above-mentioned equation in cartesian form.   [1]
2. Find the equation of the path traced by the other honeybee passing through the point (2, 4, 5).   [1]

Answer 1

a. Given, Vector form is:

`vecr = (hati + 2hatj + 3hatk) + λ(2hati + 3hatj + 4hatk)`

Put `vecr = xhati + yhatj + zhatk` in the given equation.

`xhati + yhatj + zhatk = (1 + 2λ)hati + (2 + 3λ)hatj + (3 + 4λ)hatk`

Compare the coefficient of `hati, hatj` and `hatk`.

x = 1 + 2λ, y = 2 + 3λ and z = 3 + 4λ

`(x - 1)/2 = λ, (y - 2)/3 = λ` and `(z - 3)/4 = λ`

Cartesian form is

`(x - 1)/2 = (y - 2)/3 = (z - 3)/4`

b. Equation of path traced by other honeybeе passing through (2, 4, 5) is `(x - 2)/2 = (y - 4)/3 = (z - 5)/4`