Question

The sum of the intercepts made by the plane $\vec{r}\cdot(3\hat{i}+\hat{j}+2\hat{k})=18$ on the co-ordinate axes is

• A. 30
• B. 18
• C. 33
• D. 36
• E. 27

Hint:

Shortcut Tip: To find an intercept quickly, just set the other two variables to 0. For the x-intercept of $3x+y+2z=18$, set $y=0$ and $z=0$ to get $3x=18 \implies x=6$.

Answer 1

The Correct Option is C

Concept:
The intercept form of a plane equation is $\frac{x}{A} + \frac{y}{B} + \frac{z}{C} = 1$, where $A$, $B$, and $C$ are the intercepts on the x, y, and z axes respectively. To find them, convert the vector equation to Cartesian form and divide by the constant term.

Step 1: Convert to Cartesian form.
The given vector equation is $\vec{r} \cdot (3\hat{i} + \hat{j} + 2\hat{k}) = 18$. Using $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, the Cartesian form is: $$3x + y + 2z = 18$$

Step 2: Convert to intercept form.
Divide the entire equation by 18 so that the right side becomes exactly 1: $$\frac{3x}{18} + \frac{y}{18} + \frac{2z}{18} = \frac{18}{18}$$

Step 3: Simplify the fractions.
Reduce each fraction to find the exact intercept values in the denominators: $$\frac{x}{6} + \frac{y}{18} + \frac{z}{9} = 1$$

Step 4: Identify the individual intercepts.
Comparing with $\frac{x}{A} + \frac{y}{B} + \frac{z}{C} = 1$: x-intercept ($A$) = $6$ y-intercept ($B$) = $18$ z-intercept ($C$) = $9$

Step 5: Calculate the sum of the intercepts.
Add the three intercept values together: $$\text{Sum} = 6 + 18 + 9 = 33$$ Hence the correct answer is (C) 33.