Q 5. - Calculus

Question

Let $v = \left(a, b, c\right)$ and $w = \left(\alpha , \beta , \gamma \right)$ where v and w are not parallel vectors. Explain why the planes $ax + by + cz = d$ and $\alpha x + \beta y + \gamma z = \delta$ intersect.

Step-by-Step Solution

Answer

The planes $a x+b y+c z=d$ and $\alpha x+\beta y+\gamma z=\delta$ intersect.

Step 1: Given information

Consider the vectors $\mathbf{v}=⟨a,b,c⟩$ and $\mathbf{w}=⟨\alpha ,\beta ,\gamma ⟩$ which are not parallel 

Step 2: Calculation

The objective is to explain why the planes $a x+b y+c z=d$ and $\alpha x+\beta y+\gamma z=\delta$ intersect.

Use the finding that lines in a plane either intersect or are parallel to each other to explain why the planes intersect.

The vectors $\mathbf{v}=⟨a,b,c⟩$ and $\mathbf{w}=⟨\alpha ,\beta ,\gamma ⟩$ normal to the two planes are not parallel. Therefore, they must intersect.

Therefore, the planes $a x+b y+c z=d$ and $\alpha x+\beta y+\gamma z=\delta$ intersect.