Precalculus : Parallel and Perpendicular Vectors in Two Dimensions

Example Questions

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Example Question #31 : Determine If Two Vectors Are Parallel Or Perpendicular

Find the angle between the vectors and .

Explanation:

To determine the angle between our two vectors, we can use the fact that for any 2 vectors and , where is the magnitude and is the angle between the 2 vectors, which is what we are looking for.

Working from the left, we can first find the dot product,

Now we'll find the magnitudes of the two vectors by using the Pythagorean Theorem:

take the square root of both sides

Now we can plug these values back into the equation to start solving for theta:

multiply the two numbers inside the radicals:

divide both sides by

take the inverse cosine of both sides

Example Question #32 : Determine If Two Vectors Are Parallel Or Perpendicular

Find the angle between the vectors  and . Note that the first vector is in polar form and the second is in component form.

Explanation:

To find the angle between two vectors, we can use the fact that . In order to find the dot product, we need to convert the vector to component form. This is easiest to do after drawing a quick sketch of the vector:

To find the vertical component, set up an equation involving sine, since the vertical component is the side of a right triangle across from the 20-degree angle:

evaluate the sine of 20, then multiply by 13

To find the horizontal component, set up an equation involving cosine, since the horizontal component is the side of a right triangle adjacent to the 20-degree angle:

evaluate the cosine of 20, then multiply by 13

The component form of this vector can be written as

Next, we can find the magnitude of the vector already in component form by using the Pythagorean Theorem:

take the square root of both sides

Now we have all the information that we need to solve for theta.

Find the dot product for the left side:

which we can now put back in the equation:

now divide both sides by

take invese cosine of both sides

Example Question #33 : Determine If Two Vectors Are Parallel Or Perpendicular

Find  where  and . Note that is in component form while is in polar form.

Explanation:

First, convert to component form. The easiest way to do that would be with special right triangles, knowing that for a 45-45-90 triangle the ratio of the hypotenuse to both of the legs is . So if the magnitude of the vector is , then the components must both be 3.

This could also be found by setting up and solving the equations:

and

Now that both vectors are component form, we see that we're multiplying

Example Question #34 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following best explains whether the two vectors above are perpendicular or parallel?

Neither perpendicular nor parallel, because their dot product is neither zero nor one.

Perpendicular, because their dot product is zero.

Perpendicular, because their dot product is one.

Parallel, because their dot product is one.

Parallel, because their dot product is zero.

Perpendicular, because their dot product is zero.

Explanation:

Two vectors are perpendicular if their dot product is zero, and parallel if their dot product is 1.

Take the dot product of our two vectors to find the answer:

Using our given vectors:

Thus our two vectors are perpendicular.

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