Study guide and
6 practice problems
on:
Geometric proofs with vectors
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Study Guide
Begin a geometric proof by labeling important points with as few variables as possible.
(5 problems)
The midpoint between the two vectors $\mathbf{x}$ and $\mathbf{y}$ is $\frac{\mathbf{x} + \mathbf{y}}{2}$.
(3 problems)
Related topics
Dot product
(41 problems)
Vectors
(55 problems)
Multivariable calculus
(147 problems)
Practice problems
Suppose that two opposite sides of a quadrilateral are parallel and have equal length. Show that the quadrilateral is a parallelogram.
Solution
Consider an arbitrary quadrilateral. The two blue line segments connect the midpoints of adjacent sides. Using only vector addition and multiplication by constants, show that these line segments are parallel and have the same length.
Solution
Show that the line connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side.
Solution
Show that the lines connecting any point on the semicircle of radius 1 to $(1,0)$ and $(-1,0)$ are perpendicular.
Solution
Use vectors and dot products to prove: if the diagonals of a rectangle are perpendicular, then the rectangle is a square.
Solution
Prove the parallelogram law: The sum of the squares of the lengths of both diagonals of a parallelogram equals the sum of the squares of the lengths of all four sides.
Solution