Video Transcript
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).
