Question

Let `bara` and `barb` be non-collinear vectors. If vector `barr` is coplanar with `bara` and `barb`, then show that there exist unique scalars t₁​ and t₂​ such that `barr = t_1 bara + t_2 barb`. For `barr = 2hati + 7hatj + 9hatk, bara = hati + 2hatj, barb = hatj + 3hatk`, find t₁, t₂.

Answer 1

We are given three vectors:

`barr = 2hati + 7hatj + 9hatk`

`bara = hati + 2hatj`

`barb = hatj + 3hatk`

We need to express `barr` as a linear combination of `bara` and `barb`, i.e. `barr` = t₁a + t₂b

Expanding the right-hand side,

`t_1(hati + 2hatj) + t_2(hatj + 3hatk)`

Distribute t₁ and t₂

`t_1hati + 2t_1hatj + t_2hatj + 3t_2hatk`

Grouping similar terms:

`t_1hati + (2t_1 + t_2) hatj + 3t_2hatk`

Now, comparing coefficients with `barr = 2hati + 7hatj + 9hatk`, we get the equations:

t₁ = 2 (from the `hati`-component)   ...(1)

2t₁ + t₂ = 7 (from the `hatj`-component)   ...(2)

3t₂ = 9 (from the `hatk`-component)   ...(3)

From equation (1): t₁ = 2

From equation (3): t₁ = `9/3` = 3

Substituting t₁ = 2 into equation (2):

2(2) + t₂ = 7

4 + t₂ = 7

t₂ = 7 − 4

t₂ = 3

Thus, the values of the scalars are t₁ = 2 and t₂ = 3.