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In the vector form of a line, ⃑ 𝑟 = ⃑ 𝑎 + 𝑠 ⃑ 𝑑, ⃑ 𝑎 is the position vector of a point on the line, so ( − 7, 6 ) lies on our line. Since we can rearrange this equation into the general form, we start by finding a point on the line and its slope. We recall that the equation of a line passing through ( 𝑥, 𝑦 )   and of slope 𝑚 is given by the point–slope form To apply our formula, we first need to convert the vector form into the general form. Instead, we are given the vector form of the equation of a line. In this question, we are not given the equation of our line in the general form. The perpendicular distance, 𝐷, between ( 𝑥, 𝑦 )   and 𝐿: 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 = 0 is given by To do this, we will start by recalling the following formula. We want to find the perpendicular distance between a point and a line.

This gives us the following result.Įxample 2: Finding the Distance between a Point and a Straight Line Given in Vector Form in Two Dimensionsįind the length of the perpendicular from the point ( 5, 7 ) to the straight line ⃑ 𝑟 = ( − 7, 6 ) + 𝑠 ( − 5, 7 ). We could do the same if 𝐿 was horizontal. Since these expressions are equal, the formula also holds if 𝐿 is vertical. To apply the formula, we would see 𝑎 = 𝑎, 𝑏 = 0, and 𝑐 = 𝑐, giving us If 𝐿 is vertical, then the perpendicular distance between 𝐿: 𝑎 𝑥 = − 𝑐 and 𝑃 ( 𝑥, 𝑦 )   is the absolute value of the difference in their 𝑥-coordinates:    īefore we summarize this result, it is worth noting that this formula also holds if line 𝐿 is vertical or horizontal. Substituting this into our equation for 𝐷 and simplifying gives We can do this by recalling that point 𝑅 ( 𝑥, 𝑦 )   lies on line 𝐿, so it satisfies the equation We want to find an expression for 𝐷 in terms of the coordinates of 𝑃 and the equation of line 𝐿. Substituting these into the ratio equation gives The distance between 𝑃 and 𝑅 is the absolute value of the difference in their 𝑦-coordinates: The ratio of the corresponding side lengths in similar triangles are equal, so

We call the point of intersection 𝑅, which has coordinates ( 𝑥, 𝑦 )  .

We start by dropping a vertical line from point 𝑃 to 𝐿. For example, since the line between 𝑃 and 𝑄 is perpendicular to 𝐿, we could find the equation of the line passing through 𝑃 and 𝑄 to find the coordinates of 𝑄.

There are a few options for finding this distance. We know the shortest distance between the line and the point is the perpendicular distance, so we will draw this perpendicular and label the point of intersection 𝑄. If 𝑃 lies on line 𝐿, then the distance will be zero, so let’s assume that this is not the case. If 𝐿 is vertical or horizontal, then the distance is just the horizontal/vertical distance, so we can also assume this is not the case. We want to find the shortest distance between the point 𝑃 ( 𝑥, 𝑦 )   and the line 𝐿: 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 = 0, where both 𝑎 and 𝑏 cannot both be equal to zero. The given point is called the center, \((h,k)\), and the fixed distance is called the radius, \(r\), of the circle.How To: Identifying and Finding the Shortest Distance between a Point and a Line