![]() To find the □- and □- intercepts, we set each component of the vector equation equal to zero and then solve for □. How To: Finding the □- and □-Intercepts from the Vector Form of the Equation of a Line Vector ⃑ □ is called the direction vector of the line. We can find the position vector of any point on a straight line by using a known point, □ ( □, □ ) , on the line that has a position vector, ⃑ □ , together with any nonzero vector, ⃑ □, that is parallel to the line. Let’s first explore how to find the vector equation of any line and then consider how this applies in the case of a vertical line. However, recalling that direction can also be represented using vectors, we can overcome this problem. The problem with using the slope is that it assumes the line is not vertical. As we have seen in the point–slope form, we can think of a line as a point on the line and a slope representing the direction of the line. This leads us to the vector form for the equation of a line. Finally, the standard form is difficult to find and usually involves manipulating one of the other forms of the equation of a line or being given both intercepts. Second, we cannot easily see the slope of the line in standard form. First, it is not always possible to find integer values for □, □, and □ this means we cannot write every line in the standard form. This form of the equation of a line also has some drawbacks, however. Since we can easily find both intercepts, we can use this form to sketch the line by plotting both intercepts and sketching the line that passes through these two points. Second, if both □ and □ are nonzero, we can easily find the □- and □-intercepts of the line in this form, by substituting □ = 0 and □ = 0 respectively. First, if we take □ = 0, then we can construct the equation of any vertical line.įor example, if □ = 3, □ = 0, and □ = 6, we have the line 3 □ + 0 □ = 6, that is, the vertical line □ = 2. The standard form for the equation of a line has a few advantages over the other forms noted above. ![]() If □, □, and □ are integers where □ is nonnegative, then □ □ + □ □ = □ is called the standard form for the equation of a line.Īt most one of □ or □ is allowed to be equal to zero. Recap: Standard Form of the Equation of a Line in Two Dimensions ![]()
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