The plane, passing through the points $(0,-1,2)$ and $(-1,2,1)$ and parallel to the line passing through $(5,1,-7)$ and $(1,-1,-1)$, also passes through the point :
Solution
<p>The first step is to find the normal vector to the desired plane. Since the plane is parallel to the line passing through the points (5, 1, -7) and (1, -1, -1), the direction vector of that line is also parallel to the plane. The direction vector is the difference between the coordinates of the two points, which is (5-1, 1-(-1), -7-(-1)) = (4, 2, -6).</p>
<p>Next, let's find another vector that is parallel to the plane. This vector can be obtained by taking the difference between the coordinates of the points (0, -1, 2) and (-1, 2, 1), which the plane passes through. This gives us a vector of (0-(-1), -1-2, 2-1) = (1, -3, 1).</p>
<p>The normal to the plane is perpendicular to both these vectors. It can be found by taking the cross product of the two vectors. </p>
<p>The cross product of vectors (4, 2, -6) and (1, -3, 1) is :</p>
$$
\begin{aligned}
& \vec{n}=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
4 & 2 & -6 \\
1 & -3 & 1
\end{array}\right|=\hat{i}(-16)-\hat{j}(+10)+\hat{k}(-14) \\\\
& =-16 \hat{i}-10 \hat{j}-14 \hat{k}
\end{aligned}
$$
<p>The equation of the plane can now be written in the form :</p>
<p>-16x - 10y - 14z = d</p>
<p>We can find the constant 'd' by substituting one of the points through which the plane passes, say (0, -1, 2) :</p>
<p>d = -16.0 -10.(-1) -14.2 = 10 - 28 = -18</p>
<p>So, the equation of the plane is -16x - 10y - 14z = -18.</p>
<p>Now, we substitute the given options into the equation to check which one satisfies it :</p>
<p>Option A: (-16.0 -10.5 -14.(-2) = -18) => -50 + 28 = -22 ≠ -18, so A is not correct.
<br/><br/>Option B: (-16.2 -10.0 -14.1 = -18) => -32 - 14 = -46 ≠ -18, so B is not correct.
<br/><br/>Option C: (-16.1 -10.(-2) -14.1 = -18) => -16 + 20 -14 = -10 ≠ -18, so C is not correct.
<br/><br/>Option D: (-16.(-2) -10.5 -14.0 = -18) => 32 - 50 = -18, so D is correct.</p>
<p>So, the plane also passes through the point given in Option D, which is (-2, 5, 0).</p>
About this question
Subject: Mathematics · Chapter: Three Dimensional Geometry · Topic: Direction Cosines and Ratios
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