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Find the equations of the plane the normal to which from the origin is of length 5 and which makes angles with the axes x ,y and z equal to 120 ,45 and 120 respectively?
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Find the equations of the plane the normal to which from the origin is...
Equation of a Plane with Given Normal and Angles
To find the equation of the plane with the given normal and angles, we need to follow these steps:
Step 1: Find the components of the normal vector
Let n be the normal vector of the plane. We know that the length of n is 5, so:
$$|\textbf{n}|=5$$
Since the angle between n and the x-axis is 120 degrees, we can find the x-component of n using:
$$n_x=|\textbf{n}|\cos(120^\circ)=-\frac{5}{2}$$
Similarly, we can find the y- and z-components of n:
$$n_y=|\textbf{n}|\cos(45^\circ)=\frac{5}{\sqrt{2}}$$
$$n_z=|\textbf{n}|\cos(120^\circ)=-\frac{5}{2}$$
Therefore, the normal vector of the plane is:
$$\textbf{n}=\begin{pmatrix}-\frac{5}{2}\\\frac{5}{\sqrt{2}}\\-\frac{5}{2}\end{pmatrix}$$
Step 2: Find the equation of the plane
Let r be any point on the plane, and let r0 be the position vector of a point on the plane. The equation of the plane can be written as:
$$\textbf{n}\cdot(\textbf{r}-\textbf{r}_0)=0$$
Expanding the dot product, we get:
$$-\frac{5}{2}(x-x_0)+\frac{5}{\sqrt{2}}(y-y_0)-\frac{5}{2}(z-z_0)=0$$
Dividing by 5, we get:
$$-\frac{1}{2}(x-x_0)+\frac{1}{\sqrt{2}}(y-y_0)-\frac{1}{2}(z-z_0)=0$$
This is the equation of the plane with the given normal and angles.
Community Answer
Find the equations of the plane the normal to which from the origin is...
R.n=p
r.(li^ +mj^+nk^)=5
l=n=cos120
m=cos45
therefore,
2y-√2x-√2z=10√2
r.(li^ +mj^+nk^)=5
l=n=cos120
m=cos45
therefore,
2y-√2x-√2z=10√2
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Find the equations of the plane the normal to which from the origin is of length 5 and which makes angles with the axes x ,y and z equal to 120 ,45 and 120 respectively?
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Find the equations of the plane the normal to which from the origin is of length 5 and which makes angles with the axes x ,y and z equal to 120 ,45 and 120 respectively? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Find the equations of the plane the normal to which from the origin is of length 5 and which makes angles with the axes x ,y and z equal to 120 ,45 and 120 respectively? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the equations of the plane the normal to which from the origin is of length 5 and which makes angles with the axes x ,y and z equal to 120 ,45 and 120 respectively?.
Find the equations of the plane the normal to which from the origin is of length 5 and which makes angles with the axes x ,y and z equal to 120 ,45 and 120 respectively? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Find the equations of the plane the normal to which from the origin is of length 5 and which makes angles with the axes x ,y and z equal to 120 ,45 and 120 respectively? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the equations of the plane the normal to which from the origin is of length 5 and which makes angles with the axes x ,y and z equal to 120 ,45 and 120 respectively?.
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