Question

If the line \(\overline{r}=\overline{a}+t\overline{b}\) lies on the plane \(\overline{r}\cdot\overline{n}=p\), then \(p=\):

• A. \(|\overline{a}\times\overline{b}|\)
• B. \(\overline{a}\cdot\overline{n}\)
• C. \(\overline{b}\cdot\overline{n}\)
• D. \(|\overline{a}|+|\overline{b}|\)

Hint:

If a line lies on a plane, its direction vector must always be perpendicular to the plane’s normal vector.

Answer 1

The Correct Option is B

Concept: A line lies completely on a plane if every point of the line satisfies the plane equation.

Step 1: Substitute the line into the plane equation.
\[ (\overline{a}+t\overline{b})\cdot \overline{n} = p \]

Step 2: Expand the dot product.
\[ \overline{a}\cdot\overline{n} + t(\overline{b}\cdot\overline{n}) = p \]

Step 3: Condition for all \(t\).
For the equation to hold for all \(t\), \[ \overline{b}\cdot\overline{n}=0 \] and therefore, \[ p = \overline{a}\cdot\overline{n} \]