Question

The Cartesian equation of the plane $\vec{r} = (2\hat{i} - 3\hat{j}) + \lambda(\hat{i} + 2\hat{j} - \hat{k}) + \mu(2\hat{i} + 3\hat{j} + \hat{k})$ is \dots}

• A. $5x - 4y + z = 22$
• B. $5x - 3y + z = 19$
• C. $5x - 3y - z = 19$
• D. $5x - 4y - z = 22$

Hint:

You can verify your final Cartesian equation quickly by plugging the original anchor point $(2, -3, 0)$ into it. $5(2) - 3(-3) - 0 = 10 + 9 = 19$. If it doesn't match, you've made a cross-product error!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The plane passes through point $\vec{a} = (2, -3, 0)$ and is parallel to vectors $\vec{b} = (1, 2, -1)$ and $\vec{c} = (2, 3, 1)$. The normal vector $\vec{n} = \vec{b} \times \vec{c}$.

Step 2: Formula Application:
$\vec{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -1 \\ 2 & 3 & 1 \end{vmatrix} = \hat{i}(2+3) - \hat{j}(1+2) + \hat{k}(3-4) = 5\hat{i} - 3\hat{j} - \hat{k}$.

Step 3: Explanation:
Equation: $5(x-2) - 3(y+3) - 1(z-0) = 0$ $5x - 10 - 3y - 9 - z = 0 \implies 5x - 3y - z = 19$.

Step 4: Final Answer:
The equation is $5x - 3y - z = 19$.