How to Find Vectors: An in-depth Guide with SEO Optimizations

When dealing with geometric shapes and vectors, one common task is to find the altitude of a triangle. This process can be effectively simplified using vector algebra and properties of triangles. This article will walk you through the steps to find the vector ( overrightarrow{B_1} ) such that ( overline{BB_1} ) is an altitude of triangle ( triangle ABC ).

Understanding the Geometry of Triangle ( triangle ABC )

To begin, we need to establish the coordinates and the properties of our triangle ( triangle ABC ). Given that ( AB^2 9 ), ( BC^2 50 ), and ( AC^2 59 ), it becomes evident that ( triangle ABC ) forms a right triangle with ( AC ) as the hypotenuse. This is confirmed by the Pythagorean theorem:

( AB^2 BC^2 AC^2 )

Substituting the given values:

( 9 50 59 )

This satisfies the equation, confirming the right triangle property.

Using Triangle Similarity Properties

To find the point ( B_1 ), we can use the properties of similar triangles. By drawing the altitude ( overline{BB_1} ), we create two smaller right triangles, ( triangle AB_1B ) and ( triangle CB_1B ). These triangles are similar to the larger triangle ( triangle ABC ).

The similarity ratio can be derived using the properties of similar triangles:

( frac{AB_1}{AB} frac{AB}{AC} )

And for the other leg:

( frac{B_1C}{BC} frac{BC}{AC} )

By combining these ratios, we can derive the following relationship:

( frac{AB_1}{B_1C} frac{AB^2}{BC^2} frac{9}{50} frac{3}{25} )

This indicates that ( B_1 ) divides ( AC ) in the ratio 3:25, dividing the line segment ( AC ) internally.

Determining the Point ( B_1 ) with the Section Formula

To find the exact coordinates of ( B_1 ), we can use the section formula for internal division. The section formula for a point dividing a line segment internally in the ratio ( m:n ) is given by:

( left( frac{mx_2 nx_1}{m n}, frac{my_2 ny_1}{m n} right) )

Given the ratio 3:25, and assuming the coordinates of ( A ) and ( C ) are ( (x_1, y_1) ) and ( (x_2, y_2) ) respectively, the coordinates of ( B_1 ) can be calculated as:

( left( frac{3x_2 25x_1}{28}, frac{3y_2 25y_1}{28} right) )

Assuming the coordinates are specific to the problem, the coordinates of ( B_1 ) will be:

( left( frac{47}{14}, frac{10}{7} - frac{65}{28} right) )

This simplifies to:

( left( frac{47}{14}, frac{40}{28} - frac{65}{28} right) left( frac{47}{14}, -frac{25}{28} right) )

SEO Optimizations for Best Practices

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