Question

Find the cartesian equation of the plane `vecr = (hati - hatj) + λ(hati + hatj + hatk) + µ(hati - 2hatj + 3hatk)`.

Answer 1

Given plane in vector form:

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

Step 1: Write in component form

Point on plane:

(1, –1, 0)

Direction vectors:

`veca = (1, 1, 1)`

`vecb = (1, -2, 3)`

Step 2: Find normal vector

`vecn = veca xx vecb`

`vecn = |(hati, hatj, hatk),(1, 1, 1),(1, -2, 3)|`

= `hati(1 * 3 - 1(-2)) - hatj(1 * 3 - 1 * 1) + hatk(1(-2) - 1 * 1)`

= `hati(3 + 2) - hatj(3 - 1) + hatk(-2 - 1)`

= `5hati - 2hatj - 3hatk`

So normal vector is (5, –2, –3).

Step 3: Equation of plane

Using point (1, –1, 0):

5(x – 1) – 2(y + 1) – 3(z – 0) = 0

5x – 5 – 2y – 2 – 3z = 0

5x – 2y – 3z – 7 = 0