Precalculus Examples

$f (2) = - 1$ , $f^{- 1} (9) = 4$

Step 1

Get each plane equation in standard form.

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Step 1.1

Move $2$ to the left of $f$ .

$2 f = - 1, f^{- 1} (9) = 4$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 f = - 1, \frac{1}{f} \cdot 9 = 4$

Step 1.2.2

Combine $\frac{1}{f}$ and $9$ .

$2 f = - 1, \frac{9}{f} = 4$

Step 2

To find the intersection of the line through a point $(p, q, r)$ perpendicular to plane $P_{1}$ $a x + b y + c z = d$ and plane $P_{2}$ $e x + f y + g z = h$ :

1. Find the normal vectors of plane $P_{1}$ and plane $P_{2}$ where the normal vectors are $n_{1} = ⟨ a, b, c ⟩$ and $n_{2} = ⟨ e, f, g ⟩$ . Check to see if the dot product is 0.

2. Create a set of parametric equations such that $x = p + a t$ , $y = q + b t$ , and $z = r + c t$ .

3. Substitute these equations into the equation for plane $P_{2}$ such that $e (p + a t) + f (q + b t) + g (r + c t) = h$ and solve for $t$ .

4. Using the value of $t$ , solve the parametric equations $x = p + a t$ , $y = q + b t$ , and $z = r + c t$ for $t$ to find the intersection $(x, y, z)$ .

Step 3

Find the normal vectors for each plane and determine if they are perpendicular by calculating the dot product.

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Step 3.1

$P_{1}$ is $2 f = - 1$ . Find the normal vector $n_{1} = ⟨ a, b, c ⟩$ from the plane equation of the form $a x + b y + c z = d$ .

$n_{1} = ⟨ 0, 0, 0 ⟩$

Step 3.2

$P_{2}$ is $\frac{9}{f} = 4$ . Find the normal vector $n_{2} = ⟨ e, f, g ⟩$ from the plane equation of the form $e x + f y + g z = h$ .

$n_{2} = ⟨ 0, 0, 0 ⟩$

Step 3.3

Calculate the dot product of $n_{1}$ and $n_{2}$ by summing the products of the corresponding $x$ , $y$ , and $z$ values in the normal vectors.

$0 \cdot 0 + 0 \cdot 0 + 0 \cdot 0$

Step 3.4

Simplify the dot product.

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Step 3.4.1

Remove parentheses.

$0 \cdot 0 + 0 \cdot 0 + 0 \cdot 0$

Step 3.4.2

Simplify each term.

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Step 3.4.2.1

Multiply $0$ by $0$ .

$0 + 0 \cdot 0 + 0 \cdot 0$

Step 3.4.2.2

Multiply $0$ by $0$ .

$0 + 0 + 0 \cdot 0$

Step 3.4.2.3

Multiply $0$ by $0$ .

$0 + 0 + 0$

Step 3.4.3

Simplify by adding numbers.

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Step 3.4.3.1

Add $0$ and $0$ .

$0 + 0$

Step 3.4.3.2

Add $0$ and $0$ .

$0$

Step 4

The dot product is $0$ , so the planes are perpendicular.

There is no intersection.