Question

Find the vector equation of the line passing through the points A(1, 2, 3) and B(2, 3, 4).

Answer 1

To find the vector equation of a line passing through two points A and B, we use:

The position vector of point A as the initial point.

The direction vector is given by `vec(AB)`.

The general vector equation is `vecr = veca + λvecd`, where λ is a scalar parameter.

Step-By-Step:

Step 1

Find the position vectors of points A and B:

= `A(1, 2, 3): veca = hati + 2hatj + 3hatk`

= `B(2, 3, 4): vecb = 2hati + 3hatj + 4hatk`

Step 2

Find the direction vector `vec(AB)`:

`vec(AB) = vecb - veca`

= `(2hati + 3hatj + 4hatk) - (1hati + 2hatj + 3hatk)`

= `(2 - 1)hati + (3 - 2)hatj + (4 - 3)hatk`

= `hati + hatj + hatk`

Step 3

Write the vector equation of the line:

`vecr = veca + λvec(AB)`

Substitute values:

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

The vector equation of the line passing through A(1, 2, 3) and B(2, 3, 4) is:

`vecr = (hati + 2hatj + 3hatk) + λ(hati + hatj + hatk), λ ∈ R`