Question

Find the angle between the line joining the points (3,-4,-2) and (12,2,0) and the plane 3x-y+z=1

Answer 1

Answer on Question #83676 – Math – Analytic Geometry

Question

Find the angle between the line joining the points (3, -4, -2) and (12, 2, 0) and the plane

$$3x-y+z=1$$

Solution

Equation of a line that passes through two points A(3,-4,-2) and B(12, 2, 0):

AB: $\frac{x - x_A}{x_B - x_A} = \frac{y - y_A}{y_B - y_A} = \frac{z - z_A}{z_B - z_A},$

AB: $\frac{x - 3}{12 - 3} = \frac{y + 4}{2 + 4} = \frac{z + 2}{0 + 2},$

AB: $\frac{x - 3}{9} = \frac{y + 4}{6} = \frac{z + 2}{2}.$

Then $\overrightarrow{AB}(9,6,2)$ is a direction vector of the straight line.

If the plane is given by the equation $3x-y+z=1$ , then a normal vector of the plane is $\vec{n}(3,-1,1)$ .

The angle between the straight line and the normal vector is calculated by the formula:

$$\cos \left(\overrightarrow {\mathrm {A B}}; \overrightarrow {\mathrm {n}}\right) = \frac {\overrightarrow {\mathrm {A B}} \cdot \overrightarrow {\mathrm {n}}}{\left| \overrightarrow {\mathrm {A B}} \right| \cdot \left| \overrightarrow {\mathrm {n}} \right|};$$ $$\cos \left(\overrightarrow {\mathrm {A B}}; \overrightarrow {\mathrm {n}}\right) = \frac {9 \cdot 3 + 6 \cdot (- 1) + 2 \cdot 1}{\sqrt {9 ^ {2} + 6 ^ {2} + 2 ^ {2}} \sqrt {3 ^ {2} + (- 1) ^ {2} + 1 ^ {2}}};$$ $$\cos \left(\overrightarrow {\mathrm {A B}}; \overrightarrow {\mathrm {n}}\right) = \frac {2 7 - 6 + 2}{\sqrt {8 1 + 3 6 + 4} \sqrt {9 + 1 + 1}};$$ $$\cos \left(\overrightarrow {\mathrm {A B}}; \overrightarrow {\mathrm {n}}\right) = \frac {2 3}{\sqrt {1 2 1} \sqrt {1 1}};$$ $$\cos \left(\overrightarrow {\mathrm {A B}}; \overrightarrow {\mathrm {n}}\right) = \frac {2 3}{1 1 \sqrt {1 1}};$$ $$\cos \left(\overrightarrow {\mathrm {A B}}; \overrightarrow {\mathrm {n}}\right) \approx 0. 6 3;$$

The angle between the straight line and the plane is calculated by the formula:

$$\sin \varphi = | \cos (\overrightarrow {\mathrm {A B}}; \overrightarrow {\mathrm {n}}) |;$$ $$\sin \varphi = 0. 6 3;$$ $$\varphi \approx 5 0 ^ {\prime} 5 4 ^ {\prime}.$$

Answer: $\varphi \approx 50^{\prime}54^{\prime}$

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