Mathematics · Ch 6 — Applications of Vector Algebra

Equation of a Plane Containing Two Non-Parallel Coplanar Lines

6.8.9

Equation of a Plane Containing Two Non-Parallel Coplanar Lines

Once two lines $\vec r=\vec a+s\vec b$ and $\vec r=\vec c+t\vec d$ are known to be coplanar (via §6.8.8), $\vec b\times\vec d\ne\vec 0$ tells us the plane's normal direction, and either base point tells us where the plane sits.

Parametric form. If $\vec r_0$ is the position vector of any point $P$ of that common plane, then $\vec r_0-\vec a$ (and likewise $\vec r_0-\vec c$ ) is coplanar with $\vec b,\vec d$ , so $\vec r=\vec a+t\vec b+s\vec d\qquad\text{or equivalently}\qquad \vec r=\vec c+t\vec b+s\vec d,\qquad s,t\in\mathbb R.$
Non-parametric form: $(\vec r-\vec a)\cdot(\vec b\times\vec d)=0$ (equivalently, with $\vec c$ in place of $\vec a$ ) — the same normal $\vec b\times\vec d$ used to test coplanarity in §6.8.8 now also builds the plane's equation directly. …

Figure 6.29Fig. 6.29 — Condition for coplanarity of two lines $L_1,L_2$ (directions $\vec b,\vec d$); $\vec b\times\vec d$ is perpendicular to the plane

$Fig. 6.29 — Fig. 6.29 — Condition for coplanarity of two lines $L_1,L_2$ (directions $\vec b,\vec d$); $\vec b\times\vec d$ is perpendicular to the plane$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Fig. 6.29 — Condition for coplanarity of two lines $L_1,L_2$ (directions $\vec b,\vec d$ ); $\vec b\times\vec d$ is perpendicular t …

Figure 6.30Fig. 6.30 — Angle $\theta$ between two planes equals the angle between their normals $\vec n_1,\vec n_2$

$Fig. 6.30 — Fig. 6.30 — Angle $\theta$ between two planes equals the angle between their normals $\vec n_1,\vec n_2$$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Fig. 6.30 — Angle $\theta$ between two planes equals the angle between their normals $\vec n_1,\ve …

Figure 6.31Fig. 6.31 — Angle $\theta$ between a line $\vec r=\vec a+t\vec b$ and a plane $\vec r\cdot\vec n=p$ (complement $90^\circ-\theta$ with the normal $\vec n$)

$Fig. 6.31 — Fig. 6.31 — Angle $\theta$ between a line $\vec r=\vec a+t\vec b$ and a plane $\vec r\cdot\vec n=p$ (complement $90^\circ-\theta$ with the normal $\vec n$)$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Fig. 6.31 — Angle $\theta$ between a line $\vec r=\vec a+t\vec b$ and a plane $\vec r\cdot\vec n=p$ (complement $90^\circ-\theta$ with the nor …

Figure 6.32Fig. 6.32 — Perpendicular distance $\delta$ from a point $A(\vec u)$ to a plane $\vec r\cdot\vec n=p$ (foot $F$, normal $\vec n$)

$Fig. 6.32 — Fig. 6.32 — Perpendicular distance $\delta$ from a point $A(\vec u)$ to a plane $\vec r\cdot\vec n=p$ (foot $F$, normal $\vec n$)$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Fig. 6.32 — Perpendicular distance $\delta$ from a point $A(\vec u)$ to a plane $\vec r\cdot\vec n=p$ (foot $F$ , normal …

Figure 6.33Fig. 6.33 — Line of intersection of two planes $\vec r\cdot\vec n=p$ and $\vec r\cdot\vec m=q$, parallel to $\vec n\times\vec m$

$Fig. 6.33 — Fig. 6.33 — Line of intersection of two planes $\vec r\cdot\vec n=p$ and $\vec r\cdot\vec m=q$, parallel to $\vec n\times\vec m$$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Fig. 6.33 — Line of intersection of two planes $\vec r\cdot\vec n=p$ and $\vec r\cdot\vec m=q$ , parallel to $\vec n\tim …

Figure 6.34Fig. 6.34 — A plane through the line of intersection of two given planes $\vec r\cdot\vec n_1=d_1$ and $\vec r\cdot\vec n_2=d_2$

$Fig. 6.34 — Fig. 6.34 — A plane through the line of intersection of two given planes $\vec r\cdot\vec n_1=d_1$ and $\vec r\cdot\vec n_2=d_2$$

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Fig. 6.34 — A plane through the line of intersection of two given planes $\vec r\cdot\vec n_1=d_1$ and $\vec r\cdot\vec …

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