diff --git a/docs/mcv4u7.md b/docs/mcv4u7.md index a7d7d0e..3f2f6e2 100644 --- a/docs/mcv4u7.md +++ b/docs/mcv4u7.md @@ -396,10 +396,10 @@ Much like regular multiplication, dot products are: When working with algebraic vectors, their dot products are equal to the products of their components. $$\vec{u}\bullet\vec{v}=u_xv_x+u_yv_y$$ -### Vector equations in two dimensions +### Vector line equations in two dimensions !!! definition - The **Cartesian** form of a line is of the form $Ax+By+C$. + The **Cartesian** or **scalar** form of a line is of the form $Ax+By+C$. The vector equation for a straight line solves for an unknown position vector $\vec{r}$ on the line using a known position vector $\vec{r_0}$ on the line, a direction vector parallel to the line $\vec{m}$, and the variable **parameter** $t$. It is roughly similar to $y=b+xm$. $$\vec{r}=\vec{r_0}+t\vec{m},t\in\mathbb{R}$$ @@ -408,7 +408,10 @@ The equation can be rewritten in the algebraic form to be $$[x,y]=[x_0,y_0]+t[m_1,m_2], t\in\mathbb{R}$$ The direction vector is effectively the slope of a line. -$$m=[\Delta x, \Delta y]$$ +$$\vec{m}=[\Delta x, \Delta y]$$ + +For a line in scalar form: +$$\vec{m}=[B, -A]$$ To determine if a point lies along a line defined by a vector equation, the parameter $t$ should be checked to be the same for the $x$ and $y$ coordinates of the point. @@ -432,7 +435,7 @@ If one of the **direction numbers** $m_1$ or $m_2$ is zero, the equation is rear Where $m_2=0$: $$\frac{x-x_0}{m_1},y=y_0$$ -### Vector equations in three dimensions +### Vector line equations in three dimensions There is little difference between vector equations in two or three dimensions. An additional variable is added for the third dimension. @@ -445,6 +448,17 @@ $$[x,y,z]=[x_0,y_0,z_0]+t[m_1,m_2,m_3],t\in\mathbb{R}$$ The symmetric form: $$\frac{x-x_0}{m_1}=\frac{y-y_0}{m_2}=\frac{z-z_0}{m_3}$$ +### Intersections of vector equation lines + +Two lines are parallel if their direction vectors are scalar multiples of each other. +$$\vec{m_1}=k\vec{m_2},k\in\mathbb{R}$$ + +Two lines are coincident if they are parallel and share at least one point. Otherwise, they are distinct. + +If two lines are not parallel and in two dimensions, they intersect. To solve for the point of intersection, the x and y variables in the parametric form can be equated and the parameter $t$ solved. + +In three dimensions, there is a final possibility should the lines not be parallel: the lines may be *skew*. To determine if the lines are skew, the x, y, and z variables of **two** parametric equations should be equated to their counterparts in the other vector as if they intersect. The resulting $t$ and $s$ from the first and second line respectively should be substituted into the third equation and an equality check performed. Should there not be a solution that fulfills the third equation, the lines are skew. Otherwise, they intersect. + ## Resources - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)