Calculus Examples

$y = x$ , $y = \sqrt[4]{x}$

Step 1

Get each plane equation in standard form.

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Step 1.1

Subtract $x$ from both sides of the equation.

$y - x = 0, y = \sqrt[4]{x}$

Step 1.2

Subtract $\sqrt[4]{x}$ from both sides of the equation.

$y - x = 0, y - \sqrt[4]{x} = 0$

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 $y - x = 0$ . Find the normal vector $n_{1} = ⟨ a, b, c ⟩$ from the plane equation of the form $a x + b y + c z = d$ .

$n_{1} = ⟨ - 1, 1, 0 ⟩$

Step 3.2

$P_{2}$ is $y - \sqrt[4]{x} = 0$ . Find the normal vector $n_{2} = ⟨ e, f, g ⟩$ from the plane equation of the form $e x + f y + g z = h$ .

$n_{2} = ⟨ - 1, 1, 0 ⟩$

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.

$- 1 \cdot - 1 + 1 \cdot 1 + 0 \cdot 0$

Step 3.4

Simplify the dot product.

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Step 3.4.1

Remove parentheses.

$- 1 \cdot - 1 + 1 \cdot 1 + 0 \cdot 0$

Step 3.4.2

Simplify each term.

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Step 3.4.2.1

Multiply $- 1$ by $- 1$ .

$1 + 1 \cdot 1 + 0 \cdot 0$

Step 3.4.2.2

Multiply $1$ by $1$ .

$1 + 1 + 0 \cdot 0$

Step 3.4.2.3

Multiply $0$ by $0$ .

$1 + 1 + 0$

Step 3.4.3

Simplify by adding numbers.

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Step 3.4.3.1

Add $1$ and $1$ .

$2 + 0$

Step 3.4.3.2

Add $2$ and $0$ .

$2$

Step 4

Next, build a set of parametric equations $x = p + a t$ , $y = q + b t$ , and $z = r + c t$ using the origin $(0, 0, 0)$ for the point $(p, q, r)$ and the values from the normal vector $2$ for the values of $a$ , $b$ , and $c$ . This set of parametric equations represents the line through the origin that is perpendicular to $P_{1}$ $y - x = 0$ .

$x = 0 + - 1 \cdot t$

$y = 0 + 1 \cdot t$

$z = 0 + 0 \cdot t$

Step 5

Substitute the expression for $x$ , $y$ , and $z$ into the equation for $P_{2}$ $y - \sqrt[4]{x} = 0$ .

$(0 + 1 \cdot t) - \sqrt[4]{0 - 1 \cdot t} = 0$

Step 6

Solve the equation for $t$ .

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Step 6.1

Solve for $- \sqrt[4]{0 - 1 \cdot t}$ .

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Step 6.1.1

Simplify $0 + 1 \cdot t - \sqrt[4]{0 - 1 \cdot t}$ .

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Step 6.1.1.1

Add $0$ and $1 \cdot t$ .

$1 \cdot t - \sqrt[4]{0 - 1 \cdot t} = 0$

Step 6.1.1.2

Multiply $t$ by $1$ .

$t - \sqrt[4]{0 - 1 \cdot t} = 0$

Step 6.1.2

Subtract $t$ from both sides of the equation.

$- \sqrt[4]{0 - 1 \cdot t} = - t$

Step 6.2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${(- \sqrt[4]{0 - 1 \cdot t})}^{4} = {(- t)}^{4}$

Step 6.3

Simplify each side of the equation.

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Step 6.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{0 - 1 \cdot t}$ as ${(0 - 1 \cdot t)}^{\frac{1}{4}}$ .

${(- {(0 - 1 \cdot t)}^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify ${(- {(0 - 1 \cdot t)}^{\frac{1}{4}})}^{4}$ .

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Step 6.3.2.1.1

Subtract $1 \cdot t$ from $0$ .

${(- {(- 1 \cdot t)}^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.2

Rewrite $- 1 t$ as $- t$ .

${(- {(- t)}^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.3

Apply the product rule to $- t$ .

${(- ({(- 1)}^{\frac{1}{4}} t^{\frac{1}{4}}))}^{4} = {(- t)}^{4}$

Step 6.3.2.1.4

Multiply $- 1$ by ${(- 1)}^{\frac{1}{4}}$ by adding the exponents.

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Step 6.3.2.1.4.1

Move ${(- 1)}^{\frac{1}{4}}$ .

${({(- 1)}^{\frac{1}{4}} \cdot - 1 t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.4.2

Multiply ${(- 1)}^{\frac{1}{4}}$ by $- 1$ .

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Step 6.3.2.1.4.2.1

Raise $- 1$ to the power of $1$ .

${({(- 1)}^{\frac{1}{4}} \cdot {(- 1)}^{1} t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${({(- 1)}^{\frac{1}{4} + 1} t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.4.3

Write $1$ as a fraction with a common denominator.

${({(- 1)}^{\frac{1}{4} + \frac{4}{4}} t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.4.4

Combine the numerators over the common denominator.

${({(- 1)}^{\frac{1 + 4}{4}} t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.4.5

Add $1$ and $4$ .

${({(- 1)}^{\frac{5}{4}} t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.5

Apply the product rule to ${(- 1)}^{\frac{5}{4}} t^{\frac{1}{4}}$ .

${({(- 1)}^{\frac{5}{4}})}^{4} {(t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.6

Multiply the exponents in ${({(- 1)}^{\frac{5}{4}})}^{4}$ .

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Step 6.3.2.1.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(- 1)}^{\frac{5}{4} \cdot 4} {(t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.6.2

Cancel the common factor of $4$ .

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Step 6.3.2.1.6.2.1

Cancel the common factor.

${(- 1)}^{\frac{5}{4} \cdot 4} {(t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.6.2.2

Rewrite the expression.

${(- 1)}^{5} {(t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.7

Raise $- 1$ to the power of $5$ .

$- {(t^{\frac{1}{4}})}^{4} = {(- t)}^{4}$

Step 6.3.2.1.8

Multiply the exponents in ${(t^{\frac{1}{4}})}^{4}$ .

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Step 6.3.2.1.8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- t^{\frac{1}{4} \cdot 4} = {(- t)}^{4}$

Step 6.3.2.1.8.2

Cancel the common factor of $4$ .

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Step 6.3.2.1.8.2.1

Cancel the common factor.

$- t^{\frac{1}{4} \cdot 4} = {(- t)}^{4}$

Step 6.3.2.1.8.2.2

Rewrite the expression.

$- t^{1} = {(- t)}^{4}$

Step 6.3.2.1.9

Simplify.

$- t = {(- t)}^{4}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Simplify ${(- t)}^{4}$ .

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Step 6.3.3.1.1

Apply the product rule to $- t$ .

$- t = {(- 1)}^{4} t^{4}$

Step 6.3.3.1.2

Raise $- 1$ to the power of $4$ .

$- t = 1 t^{4}$

Step 6.3.3.1.3

Multiply $t^{4}$ by $1$ .

$- t = t^{4}$

Step 6.4

Solve for $t$ .

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Step 6.4.1

Subtract $t^{4}$ from both sides of the equation.

$- t - t^{4} = 0$

Step 6.4.2

Factor the left side of the equation.

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Step 6.4.2.1

Factor $- t$ out of $- t - t^{4}$ .

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Step 6.4.2.1.1

Reorder $- t$ and $- t^{4}$ .

$- t^{4} - t = 0$

Step 6.4.2.1.2

Factor $- t$ out of $- t^{4}$ .

$- t \cdot t^{3} - t = 0$

Step 6.4.2.1.3

Factor $- t$ out of $- t$ .

$- t \cdot t^{3} - t \cdot 1 = 0$

Step 6.4.2.1.4

Factor $- t$ out of $- t (t^{3}) - t (1)$ .

$- t (t^{3} + 1) = 0$

Step 6.4.2.2

Rewrite $1$ as $1^{3}$ .

$- t (t^{3} + 1^{3}) = 0$

Step 6.4.2.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = t$ and $b = 1$ .

$- t ((t + 1) (t^{2} - t \cdot 1 + 1^{2})) = 0$

Step 6.4.2.4

Factor.

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Step 6.4.2.4.1

Simplify.

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Step 6.4.2.4.1.1

Multiply $- 1$ by $1$ .

$- t ((t + 1) (t^{2} - t + 1^{2})) = 0$

Step 6.4.2.4.1.2

One to any power is one.

$- t ((t + 1) (t^{2} - t + 1)) = 0$

Step 6.4.2.4.2

Remove unnecessary parentheses.

$- t (t + 1) (t^{2} - t + 1) = 0$

Step 6.4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$t = 0$

$t + 1 = 0$

$t^{2} - t + 1 = 0$

Step 6.4.4

Set $t$ equal to $0$ .

$t = 0$

Step 6.4.5

Set $t + 1$ equal to $0$ and solve for $t$ .

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Step 6.4.5.1

Set $t + 1$ equal to $0$ .

$t + 1 = 0$

Step 6.4.5.2

Subtract $1$ from both sides of the equation.

$t = - 1$

Step 6.4.6

Set $t^{2} - t + 1$ equal to $0$ and solve for $t$ .

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Step 6.4.6.1

Set $t^{2} - t + 1$ equal to $0$ .

$t^{2} - t + 1 = 0$

Step 6.4.6.2

Solve $t^{2} - t + 1 = 0$ for $t$ .

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Step 6.4.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.4.6.2.2

Substitute the values $a = 1$ , $b = - 1$ , and $c = 1$ into the quadratic formula and solve for $t$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 6.4.6.2.3

Simplify.

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Step 6.4.6.2.3.1

Simplify the numerator.

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Step 6.4.6.2.3.1.1

Raise $- 1$ to the power of $2$ .

$t = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.4.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 6.4.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.4.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

$t = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 6.4.6.2.3.1.3

Subtract $4$ from $1$ .

$t = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.4.6.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$t = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.4.6.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$t = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.4.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$t = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.4.6.2.3.2

Multiply $2$ by $1$ .

$t = \frac{1 \pm i \sqrt{3}}{2}$

Step 6.4.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.4.6.2.4.1

Simplify the numerator.

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Step 6.4.6.2.4.1.1

Raise $- 1$ to the power of $2$ .

$t = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.4.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 6.4.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.4.6.2.4.1.2.2

Multiply $- 4$ by $1$ .

$t = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 6.4.6.2.4.1.3

Subtract $4$ from $1$ .

$t = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.4.6.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$t = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.4.6.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$t = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.4.6.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$t = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.4.6.2.4.2

Multiply $2$ by $1$ .

$t = \frac{1 \pm i \sqrt{3}}{2}$

Step 6.4.6.2.4.3

Change the $\pm$ to $+$ .

$t = \frac{1 + i \sqrt{3}}{2}$

Step 6.4.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.4.6.2.5.1

Simplify the numerator.

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Step 6.4.6.2.5.1.1

Raise $- 1$ to the power of $2$ .

$t = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.4.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 6.4.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.4.6.2.5.1.2.2

Multiply $- 4$ by $1$ .

$t = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 6.4.6.2.5.1.3

Subtract $4$ from $1$ .

$t = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.4.6.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$t = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.4.6.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$t = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.4.6.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$t = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.4.6.2.5.2

Multiply $2$ by $1$ .

$t = \frac{1 \pm i \sqrt{3}}{2}$

Step 6.4.6.2.5.3

Change the $\pm$ to $-$ .

$t = \frac{1 - i \sqrt{3}}{2}$

Step 6.4.6.2.6

The final answer is the combination of both solutions.

$t = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 6.4.7

The final solution is all the values that make $- t (t + 1) (t^{2} - t + 1) = 0$ true.

$t = 0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 7

Solve the parametric equations for $x$ , $y$ , and $z$ using the value of $t$ .

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Step 7.1

Solve the equation for $x$ .

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Step 7.1.1

Remove parentheses.

$x = 0 - 1 \cdot (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$

Step 7.1.2

Simplify $0 - 1 \cdot (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$ .

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Step 7.1.2.1

Simplify each term.

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Step 7.1.2.1.1

Multiply $- 1$ by each element of the matrix.

$x = 0 + (- 0, - - 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2})$

Step 7.1.2.1.2

Simplify each element in the matrix.

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Step 7.1.2.1.2.1

Multiply $- 1$ by $0$ .

$x = 0 + (0, - - 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2})$

Step 7.1.2.1.2.2

Multiply $- 1$ by $- 1$ .

$x = 0 + (0, 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2})$

Step 7.1.2.2

Add $0$ and $(0, 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2})$ .

$x = (0, 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2})$

Step 7.2

Solve the equation for $y$ .

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Step 7.2.1

Remove parentheses.

$y = 0 + 1 \cdot (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$

Step 7.2.2

Simplify $0 + 1 \cdot (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$ .

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Step 7.2.2.1

Multiply $(0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$ by $1$ .

$y = 0 + (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$

Step 7.2.2.2

Add $0$ and $(0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$ .

$y = (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$

Step 7.3

Solve the equation for $z$ .

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Step 7.3.1

Remove parentheses.

$z = 0 + 0 \cdot (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$

Step 7.3.2

Simplify $0 + 0 \cdot (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$ .

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Step 7.3.2.1

Simplify each term.

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Step 7.3.2.1.1

Multiply $0$ by each element of the matrix.

$z = 0 + (0 \cdot 0, 0 \cdot - 1, 0 \frac{1 + i \sqrt{3}}{2}, 0 \frac{1 - i \sqrt{3}}{2})$

Step 7.3.2.1.2

Simplify each element in the matrix.

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Step 7.3.2.1.2.1

Multiply $0$ by $0$ .

$z = 0 + (0, 0 \cdot - 1, 0 \frac{1 + i \sqrt{3}}{2}, 0 \frac{1 - i \sqrt{3}}{2})$

Step 7.3.2.1.2.2

Multiply $0$ by $- 1$ .

$z = 0 + (0, 0, 0 \frac{1 + i \sqrt{3}}{2}, 0 \frac{1 - i \sqrt{3}}{2})$

Step 7.3.2.1.2.3

Multiply $0$ by $\frac{1 + i \sqrt{3}}{2}$ .

$z = 0 + (0, 0, 0, 0 \frac{1 - i \sqrt{3}}{2})$

Step 7.3.2.1.2.4

Multiply $0$ by $\frac{1 - i \sqrt{3}}{2}$ .

$z = 0 + (0, 0, 0, 0)$

Step 7.3.2.2

Add $0$ and $(0, 0, 0, 0)$ .

$z = (0, 0, 0, 0)$

Step 7.4

The solved parametric equations for $x$ , $y$ , and $z$ .

$x = (0, 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2})$

$y = (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$

$z = (0, 0, 0, 0)$

$x = (0, 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2})$

$y = (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2})$

$z = (0, 0, 0, 0)$

Step 8

Using the values calculated for $x$ , $y$ , and $z$ , the intersection point is found to be $((0, 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2}), (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}), (0, 0, 0, 0))$ .

$((0, 1, - \frac{1 + i \sqrt{3}}{2}, - \frac{1 - i \sqrt{3}}{2}), (0, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}), (0, 0, 0, 0))$