Question Video: Determining in Vector Form the Equation of the Median Drawn from the Point of the Vertex of a Triangle | Nagwa Question Video: Determining in Vector Form the Equation of the Median Drawn from the Point of the Vertex of a Triangle | Nagwa

Question Video: Determining in Vector Form the Equation of the Median Drawn from the Point of the Vertex of a Triangle Mathematics • Third Year of Secondary School

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The points 𝐴 (βˆ’8, βˆ’9, βˆ’2), 𝐡 (0, βˆ’7, 6), and 𝐢 (βˆ’8, βˆ’1, βˆ’4) form a triangle. Determine in vector form, the equation of the median drawn from 𝐢.

03:19

Video Transcript

The points 𝐴 negative eight, negative nine, negative two; 𝐡 zero, negative seven, six; and 𝐢 negative eight, negative one, negative four form a triangle. Determine in vector form the equation of the median drawn from 𝐢.

In this question, we have three points 𝐴, 𝐡, and 𝐢, which are given in three-dimensional space. These three points we’re told form a triangle. We’re told that there is a median drawn from 𝐢, and so it would be useful to recall that a median is a line segment joining a vertex to the midpoint of the opposite side. For example, if we drew this two-dimensional triangle 𝐴𝐡𝐢, the median from 𝐢 would look like this.

Perhaps the best way to begin this question is to see if we can find the point that is the midpoint of 𝐴𝐡. Let’s define this with the letter 𝑀. The formula to find the midpoint of two points in space is very similar to that which we might use for two coordinates in two-dimensional space. To find the midpoint 𝑀 of π‘₯ one, 𝑦 one, 𝑧 one and π‘₯ two, 𝑦 two, 𝑧 two, we have that 𝑀 is equal to π‘₯ one plus π‘₯ two over two, 𝑦 one plus 𝑦 two over two, 𝑧 one plus 𝑧 two over two. When we fill our values into this formula, we need to make sure we’re using the values for 𝐴 and 𝐡 as, after all, we need to find the midpoint of 𝐴𝐡.

Note that when we’re plugging in our values, it doesn’t matter which point we use with our π‘₯ one, 𝑦 one, 𝑧 one values or the π‘₯ two, 𝑦 two, 𝑧 two values. So we have that the midpoint 𝑀 is equal to negative eight plus zero over two, negative nine plus negative seven over two, and negative two plus six over two. Simplifying this, we have that 𝑀 is equal to negative four, negative eight, two. We can now clear some space so we can begin to think about the vector form of the equation of this median. The vector form of an equation can be written in the form 𝐫 equals 𝐫 sub zero plus 𝑑𝐯, where 𝐫 is the position vector of a general point on the line, 𝐫 sub zero is the position vector of a given point on the line, and 𝐯 is the direction vector. 𝑑 is a scalar multiple.

Let’s think about what would happen if we model these three points in three-dimensional space. We’d have the triangle 𝐴𝐡𝐢 and the median, which would be the line segment of 𝐢𝑀. So when it comes to writing the median in vector form, the position vector can be the point 𝐢. But we still need to work out the direction vector of π‚πŒ. To find the vector π‚πŒ, we subtract the starting point 𝐢 from the terminal point 𝑀. So we have negative four subtract negative eight, negative eight subtract negative one, and two subtract negative four. Simplifying this, we have that vector π‚πŒ is equal to four, negative seven, six.

Now we have all the information that we need to plug in to the vector form of the line. 𝐫 sub zero will be the position vector representing point 𝐢. Vector 𝐯 will be represented by the vector π‚πŒ. Therefore, the answer for the equation of the median from 𝐢 is 𝐫 equals negative eight, negative one, negative four plus 𝑑 four, negative seven, six.

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