Video: Parallel and Perpendicular Vectors in 2D

In this video, we will learn how to recognize parallel and perpendicular vectors in 2D.

17:01

Video Transcript

In this video, we will learn how to recognize parallel and perpendicular vectors in two dimensions. Remember that if we’re thinking about lines, then two parallel lines will be two lines that never cross. Two perpendicular lines would be two lines which cross at right angles. So let’s begin by reminding ourselves about vectors and thinking about how we might recognize parallel vectors.

Let’s start by looking at this vector 𝐮. We could write this vector in terms of its horizontal and vertical components as the vector two, one. We can now introduce some other vectors, let’s say this vector 𝐚, which we can write in terms of the horizontal and vertical components as four, two. We can even draw another vector, vector 𝐛, which looks similar to vector 𝐮, only this time it’s going in the opposite direction. We could even draw in one final vector, vector 𝐜, written as negative six, negative three.

What do we notice about the four vectors that we’ve drawn? Well, they’re all parallel. All of these vectors are scalar multiples of each other. For example, if we multiplied vector 𝐮 by two, we would get vector 𝐚. Even if we think about how we would go from vector 𝐮 to vector 𝐛, we could multiply vector 𝐮 by negative one to get vector 𝐛. We can write a formal definition of parallel vectors in the following way. We can say that for any two vectors 𝐮 and 𝐯, they are parallel if 𝐮 is equal to 𝑘 times 𝐯 for some scalar 𝑘, where 𝑘 is not equal to zero. So, if we look at our diagram on vectors 𝐛 and 𝐜, we could write that vector 𝐛 is equal to 𝑘 times vector 𝐜. We could even work out that 𝑘 in this case would be one-third.

However, if we didn’t have a diagram and we didn’t have any information about the lengths of 𝐛 or 𝐜, then we would know that they must be parallel because they’re scalar multiples of each other. We’re now going to look at some questions. And in the first one, we’ll identify if two given vectors are parallel or not.

True or false: vectors 𝐀 equals two, one and 𝐁 equals six, three are parallel. Option (A) true or option (B) false.

In this question, we’re given information about two vectors 𝐀 and 𝐁, and we’re asked to work out if these two vectors are parallel. We can recall that two vectors are parallel if they are scalar multiples of each other. If vectors 𝐀 and 𝐁 are parallel, then we could write that vector 𝐀 is equal to 𝑘 times vector 𝐁 for some scalar quantity 𝑘, where 𝑘 is not equal to zero. Given the information about vectors 𝐀 and 𝐁, then we can fill these into this equation and check if it’s valid. You might notice that vector 𝐁 is a nice multiple of vector 𝐀. But it’s worth checking the 𝑥- and 𝑦-components separately.

Evaluating the 𝑥-components, we’d have two is equal to six 𝑘. Dividing both sides by six would give us two-sixths is equal to 𝑘. Simplifying would give us a third equals 𝑘 or 𝑘 is equal to one-third. In order for these vectors to be parallel, then when we evaluate the 𝑦-components, we need to get the same value of 𝑘. So, let’s check. When we evaluate the 𝑦-components, we get that one is equal to three 𝑘. Dividing both sides by three, we’d get that one-third is equal to 𝑘 which, of course, means that 𝑘 is equal to one-third. We can then say that vector 𝐀 does equal 𝑘 times vector 𝐁 because we can write that vector 𝐀 is equal to one-third times vector 𝐁. We could therefore give the answer true since the statement “vectors 𝐀 and 𝐁 are parallel” is true.

One alternative way to answer this question would be to represent these vectors on a diagram. Vector 𝐀 is given as two, one, and vector 𝐁 is given as six, three. We can see visually that vectors 𝐀 and 𝐁 are parallel, confirming the answer that the statement is true.

We’ll now have a look at how we identify perpendicular vectors. We say that two vectors 𝐮 equals 𝑥 one, 𝑦 one and 𝐯 equals 𝑥 two, 𝑦 two are perpendicular if the dot product of 𝐮 and 𝐯 is equal to zero. We recall that we can find the dot product of vectors 𝐮 and 𝐯 by finding the product of 𝑥 one and 𝑥 two and adding it to the product of 𝑦 one and 𝑦 two. So, let’s see how we can use this definition to help us to identify perpendicular vectors.

Which of the following vector pairs are perpendicular? Option (A) vector two, zero and vector three, negative six. Option (B) vector one, four and vector two, eight. Option (C) vector zero, seven and vector zero, nine. Or option (D) vector three, zero and vector zero, six.

Let’s begin by recalling how we identify if vectors are perpendicular. If we have two vectors 𝐮 and 𝐯 which are perpendicular, then the dot product of vector 𝐮 and 𝐯 is equal to zero. We can remind ourselves of how to find the dot product by saying that if vector 𝐮 is given by 𝑥 one, 𝑦 one and vector 𝐯 is given by 𝑥 two, 𝑦 two, then the dot product 𝐮𝐯 is equal to 𝑥 one times 𝑥 two plus 𝑦 one times 𝑦 two. So, in each of these options, (A) to (D), we’ll work out this dot product, and if it’s equal to zero, then the vector pair will be perpendicular. So, let’s start with the vector pair given in option (A). And we can identify 𝑥 one, 𝑦 one, 𝑥 two, and 𝑦 two values, although it doesn’t matter which vector we choose for each of the 𝑥 one and 𝑦 one values.

To find the dot product then, we’ll have two times three plus zero times negative six. Evaluating this, two times three gives us six. And be careful because, of course, zero times negative six is zero. Six plus zero simplifies to six. So, did we calculate the dot product equal to zero? No, we did not. Therefore, the vector pair given in option (A) are not perpendicular. We can follow the same process then for the vectors given in option (B). When we calculate the dot product here, we have one times two plus four times eight. One times two is two, and four times eight is 32. Adding these together gives us the value of 34. As this dot product is not equal to zero, then the vectors given in option (B) are not perpendicular.

Applying the same method for the vectors given in option (C), we’re multiplying zero by zero and adding it to seven times nine, which gives us 63. 63 is not equal to zero, so the vectors in option (C) are not perpendicular. Finally, in option (D), each of the products of 𝑥 one, 𝑥 two and 𝑦 one, 𝑦 two will give us zero. So, when we add these together, we get zero. As we have found a dot product of vectors which is equal to zero, then these two vectors given in option (D) are perpendicular. Therefore, we can give the answer that vector three, zero and vector zero, six are perpendicular.

We can confirm this by drawing these two vectors. The vector three, zero could be represented by a line going three units to the right and zero units up. Vector zero, six could be represented by a line which goes zero units horizontally and six units upwards. The first vector is a horizontal line, and the second vector is a vertical line, indicating that these two vectors are indeed perpendicular and so confirming that the vector pair which is perpendicular are those given in option (D).

We’ll now have a look at a few more questions involving parallel and perpendicular vectors. In the next question, we’ll need to find a missing value in a pair of parallel vectors.

If vector 𝐀 is equal to ℎ, ℎ plus two and vector 𝐁 is equal to three ℎ, four ℎ minus one, then one of the values of ℎ that makes vector 𝐀 parallel to vector 𝐁 is what. Option (A) seven, option (B) five, option (C) negative five, or option (D) negative seven.

In this question, we’re given two vectors 𝐀 and 𝐁. Their components include this unknown value of ℎ. We need to find one of the values of ℎ such that vector 𝐀 and 𝐁 will be parallel. So, let’s begin by recalling what it means to have two parallel vectors. In this case, as we have vector 𝐀 and 𝐁, we would say that they’re parallel if we can write that vector 𝐀 is equal to 𝑘 times vector 𝐁 for some scalar quantity 𝑘, where 𝑘 is not equal to zero. So, let’s set our two vectors as parallel vectors by saying that vector 𝐀 is equal to 𝑘 times vector 𝐁. We can then fill in the values of the vectors that we’re given. Vector 𝐀 is ℎ, ℎ plus two, and vector 𝐁 is three ℎ, four ℎ minus one.

On the right-hand side, we can multiply 𝑘 by each of the 𝑥- and 𝑦-components. We can then evaluate the 𝑥-components. So we would have ℎ is equal to three 𝑘ℎ. Dividing both sides by ℎ, we have that one is equal to three 𝑘. Dividing three by three, we have one-third is equal to 𝑘 or 𝑘 is equal to one-third. Next, let’s consider the 𝑦-components. We can write that ℎ plus two is equal to 𝑘 times four ℎ minus one. We have already established that 𝑘 is equal to one-third, so let’s substitute this into the equation. We then have that ℎ plus two is equal to one-third times four ℎ minus one. Now let’s expand the parentheses on the right-hand side. One-third multiplied by four ℎ will give us four-thirds ℎ, and one-third multiplied by negative one gives us negative one-third.

We can then subtract ℎ from both sides, remembering that if we have four-thirds ℎ and we take away ℎ, that will leave us with one-third ℎ. We can then add one-third to both sides, remembering that two and one-third is equivalent to seven-thirds. Finally, we can multiply by three on both sides of this equation, which leaves us with seven is equal to one ℎ or simply seven is equal to ℎ. Therefore, one of the values of ℎ that makes 𝐀 parallel to 𝐁 is seven. So, the answer is that given in option (A) seven. We can check this by plugging the value of seven into vectors 𝐀 and 𝐁. When ℎ is equal to seven, vector 𝐀 will be seven, and seven plus two is nine. For vector 𝐁, when ℎ is equal to seven, three times seven is 21 and four times seven is 28; subtract one will give us 27.

So, are vectors 𝐀 and 𝐁 scalar multiples? Yes, they are because we could write that vector 𝐀 is equal to one-third of vector 𝐁. So, we’ve confirmed that when ℎ is seven, vectors 𝐀 and 𝐁 are parallel. But let’s check if any of the options given in (B), (C), or (D) for the value of ℎ will also make the vectors 𝐀 and 𝐁 parallel. This time, we’ll take the value given in the answer option for the value of ℎ and then check if the vector 𝐀 is parallel to vector 𝐁.

So, for option (B), we’re checking the value of ℎ is equal to five. And vector 𝐀 will therefore be five, and five plus two is seven. And vector 𝐁 will be three times five, which is 15, and four times five is 20 minus one is 19. So, is there a scalar value 𝑘 for which the vector five, seven is equal to 𝑘 times the vector 15, 19? Well, no, there’s not. If we look at the 𝑥-components, we’ll have five is equal to 15𝑘. That would mean that 𝑘 is equal to one-third. However, 𝑘 is equal to one-third would not fit the equation seven is equal to 𝑘 times 19. So, that means when ℎ is equal to five, these two vectors five, seven and 15, 19 are not parallel. So, we can eliminate option (B).

Let’s check option (C). This time we’re checking the value of ℎ is equal to negative five. So, vector 𝐀 will be negative five, negative three, and vector 𝐁 will be negative 15, negative 21. Once again, there is no value of 𝑘 for which vector 𝐀 is equal to 𝑘 times vector 𝐁. So, we can eliminate option (C) because when ℎ is equal to negative five, these two vectors are not parallel. Finally then, in option (D), we’re checking the value of ℎ is equal to negative seven. When we plug this value into the vectors 𝐀 and 𝐁, once again, there is no value of 𝑘 which makes negative seven, negative five and negative 21, negative 29 parallel. We have therefore eliminated options (B), (C), and (D), leaving us with the value of seven.

We’ll now look at one final question.

Fill in the blank: vectors 𝐀 equals one, two and vector 𝐁 equals negative two, one are what.

The first thing we might consider checking is if vectors 𝐀 and 𝐁 are parallel. We can recall that any two vectors 𝐀 and 𝐁 are parallel if we can write that vector 𝐀 is equal to 𝑘 times vector 𝐁 for any scalar 𝑘, where 𝑘 is not equal to zero. So, let’s check if this true for the given vectors. Is there a value 𝑘 for which the vector one, two is equal to 𝑘 times negative two, one? We can simplify the vector on the right-hand side as negative two 𝑘, 𝑘. We can then evaluate the 𝑥- and 𝑦-components separately, so the 𝑥-components would give us one is equal to negative two 𝑘. Dividing this equation by negative two on both sides would give us that negative one-half is equal to 𝑘. So, 𝑘 is equal to negative one-half.

Let’s have a look at the 𝑦-components. This gives us the equation two is equal to 𝑘 or, alternatively, 𝑘 is equal to two. However, we have two different values of 𝑘, which means that there is no value of 𝑘 for which vector 𝐀 is equal to 𝑘 times vector 𝐁. This means that these two vectors are not parallel, so let’s see what else they might be. Let’s check if they’re perpendicular. We recall that if the dot product of two vectors is equal to zero, then those vectors are perpendicular. For any two vectors 𝐮 is equal to 𝑥 one, 𝑦 one and 𝐯 is equal to 𝑥 two, 𝑦 two, then the dot product 𝐮𝐯 is equal to the product 𝑥 one 𝑥 two plus the product 𝑦 one 𝑦 two.

To calculate the dot product of our two vectors 𝐀 and 𝐁, then we work out one times negative two plus two times one. And we can simplify this to negative two plus two which is equal to zero. For any two vectors 𝐮 and 𝐯, they are perpendicular if the dot product 𝐮 dot 𝐯 is equal to zero. We have established that the dot product of vectors 𝐀 and 𝐁 here is equal to zero. This means that we can fill in the blank. Vectors 𝐀 is equal to one, two and 𝐁 is equal to negative two, one are perpendicular.

We can now summarize the key points of this video. We saw that two vectors 𝐮 and 𝐯 are parallel if vector 𝐮 is equal to 𝑘 times vector 𝐯 for some scalar quantity 𝑘, where 𝑘 is not equal to zero. Finally, we saw that vectors 𝐮 and 𝐯 are perpendicular if the dot product of vectors 𝐮 and 𝐯 is equal to zero.

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