We have already reached our solution, but, to illustrate the method, let us complete the proof that they are not perpendicular. This tells us that our vectors are in fact parallel. We obtainīy solving any of these equations, we will get the same value of □ = 1 8. We can form three equations by equating the components of these vectors. We can check whether this is true by trying to find a value of □. If they are parallel, then it would be true that Let us start by checking whether they are parallel. If neither of these conditions are met, then the vectors are neither parallel nor perpendicular to one another. ⃑ □ = □ ⃑ □, where □ is a nonzero real constant. Let us start by recalling the conditions under which these vectors are either parallel or perpendicular. Two vectors are parallel, perpendicular, or otherwise. In the next example, we will be determining whether two vectors are parallel, perpendicular, or neither.Įxample 4: Identifying Whether Two Vectors are Parallel, Perpendicular, or Otherwise Since this result is zero, this helps to confirm our solution: D. We can quickly check the vector E, just to make sure that this vector is perpendicular to the line: So, our solution to the question is that the vector that is not perpendicular to the line is D, ( 1, − 2, 3 ). Since this dot product is nonzero, ( 1, − 2, 3 ) and ( 2, − 3, 5 ) are not perpendicular. Since this is zero, the answer is also not B.ĭotting the vector in C with ( 2, − 3, 5 ), we obtain Therefore, ( 1 0, 1 0, 2 ) is perpendicular to ( 2, − 3, 5 ). We just need to find which of the vectors does not give zero when dotted with ( 2, − 3, 5 ). In order for two vectors to be perpendicular to one another, it must be true that their dot product is equal to zero. Which of the following vectors is not perpendicular to the line whose direction vector ⃑ □ is ( 2, − 3, 5 )? In the next example, we will look at how we can identify perpendicular vectors.Įxample 3: Determining the Vector That is Not Perpendicular to the Given Line This confirms that our solution is correct. If we substitute in the values we obtained, we get We can check our solution by checking that the ratios of corresponding components of the two vectors are equal.įor two parallel vectors, it should be true that We have now reached our solution, which is that the values of □ and □ that make the vectors parallel are Now, we simply need to substitute this value into the other two equations and solve for the missing values. We can solve the first of these equations to find □. Where □ is just a constant that can be found.īy equating the coefficients of each of the vector components, we end up with three equations: In order to solve this problem, we can use the fact that when two vectors are parallel to one another, they are scalar multiples of one another. Example 2: Finding a Missing Value Using a Pair of Parallel Vectorįind the values of □ and □ so that vector
0 Comments
Leave a Reply. |