Mathematics · Ch 6 — Applications of Vector Algebra

The Coordinates of the Image of a Point in a Plane

6.9.1

The Coordinates of the Image of a Point in a Plane

The image formula of §6.9 restates cleanly in pure coordinates. Let $(a_1,a_2,a_3)$ be the given point (so $\vec u=a_1\hat i+a_2\hat j+a_3\hat k$ ) and let $ax+by+cz=d$ be the given plane (so $\vec n=a\hat i+b\hat j+c\hat k$ , $p=d$ ). Writing the image as $\vec v=v_1\hat i+v_2\hat j+v_3\hat k$ , matching $\hat i,\hat j,\hat k$ -components of $\vec v=\vec u+2\alpha\vec n$ gives

$v_1=a_1+2a\alpha,\qquad v_2=a_2+2b\alpha,\qquad v_3=a_3+2c\alpha,\qquad\text{where}\qquad \alpha=\frac{d-(aa_1+ba_2+ca_3)}{a^2+b^2+c^2}.$ …

Figure 6.36Fig. 6.36 — Point $M$ where a line $\vec r=\vec a+t\vec b$ (direction $\vec b$) meets a plane $\vec r\cdot\vec n=p$

$Fig. 6.36 — Fig. 6.36 — Point $M$ where a line $\vec r=\vec a+t\vec b$ (direction $\vec b$) meets a plane $\vec r\cdot\vec n=p$$

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. 6.36 — Point $M$ where a line $\vec r=\vec a+t\vec b$ (direction $\vec b$ ) meets a plane $\vec r\cdot\v …

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