3. The Cross Product of Two Vectors - 10. Analytic Geometry in Three Dimensions - Precalculus with Limits: A Graphing Approach, Sixth Edition

We want to complete the sentence.

To find a vector in space that is orthogonal to two given vectors, find the of the two vectors.

First, let's recall the definition of orthogonal vectors.

Orthogonal Vectors
If the dot product of two vectors u and v is equal to zero, then the angle between these vectors is 90^(∘) and the vectors are orthogonal.

Let's consider some three-dimensional vectors u and v. There are infinitely many vectors that are orthogonal to both of these vectors.

In this chapter, we learn how to calculate the cross product of two vectors u = and v = .

Cross Product of Two Vectors
u* v = (u_2v_3 - u_3v_2)i - (u_1v_3 - u_3v_1)j + (u_1v_2 - u_2v_1)k

The cross product of u and v is an example of a vector that is orthogonal to both u and v. We can show an example on the graph. Let's find the cross product of unit vectors u = <1,0,0> and v = <0,1,0>.

u* v = (u_2v_3 - u_3v_2)i - (u_1v_3 - u_3v_1)j + (u_1v_2 - u_2v_1)k

SubstituteValues

Substitute values

u* v = (0* 0 - 0* 1)i - (1* 0 - 0* 0)j + (1* 1 - 0* 0)k

Multiply

u* v = 0* i - 0* j + k

Substitute

u* v = 0* <1,0,0> - 0* <0,1,0> + <0,0,1>

Multiply

u* v =<0,0,0> - <0,0,0> + <0,0,1>

⟨ a,b⟩±⟨ c,d⟩=⟨ a ± c,b ± d⟩

u* v =<0,0,1>

Let's draw the vectors u, v, and u* v on the coordinate system.

We can see that the cross product of vectors <1,0,0> and <0,1,0> is orthogonal to both of these vectors. To sum up, the cross product of two vectors is orthogonal to both vectors. Finally, let's fill in the blank.

To find a vector in space that is orthogonal to two given vectors, find the cross product of the two vectors.

Exercise 1 Page 731

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