Video Transcript
If vector 𝐀 is equal to 11𝐢 plus
10𝐣, vector 𝐁 is equal to negative five 𝐢 plus 14𝐣, vector 𝐌 is equal to vector
𝐀 plus vector 𝐁, and vector 𝐍 is equal to vector 𝐀 minus vector 𝐁, then which
one of the following statements is true? Is it (A) vector 𝐌 is equal to
vector 𝐍? (B) The magnitude of vector 𝐌 is
equal to the magnitude of vector 𝐍. Is it (C) vector 𝐌 is parallel to
vector 𝐍 or option (D) vector 𝐌 is perpendicular to vector 𝐍?
In this question, we are given
vector 𝐀 and vector 𝐁 in terms of the unit vectors 𝐢 and 𝐣. We are told that vector 𝐌 is equal
to 𝐀 plus 𝐁 and vector 𝐍 is equal to 𝐀 minus 𝐁. We will begin by calculating these
two vectors. Vector 𝐌 is equal to 11𝐢 plus
10𝐣 plus negative five 𝐢 plus 14𝐣. This can be simplified by adding
the 𝐢- and 𝐣-components separately. 11 plus negative five is equal to
six, and 10 plus 14 is equal to 24. Vector 𝐌 is, therefore, equal to
six 𝐢 plus 24𝐣.
We can calculate vector 𝐍 by
subtracting negative five 𝐢 plus 14𝐣 from 11𝐢 plus 10𝐣. Once again, we’ll subtract the 𝐢-
and 𝐣-components separately. 11 minus negative five is the same
as 11 plus five, which equals 16. Therefore, the 𝐢-component equals
16. 10 minus 14 is equal to negative
four. Therefore, vector 𝐍 is equal to
16𝐢 minus four 𝐣.
Option (A) said that vector 𝐌 was
equal to vector 𝐍. Clearly, this is not the case as
six 𝐢 plus 24𝐣 is not equal to 16𝐢 minus four 𝐣. The second option said that the
magnitude of vector 𝐌 was equal to the magnitude of vector 𝐍. Recalling that the magnitude of any
vector is equal to the square root of the sum of the squares of its individual
components. The magnitude of vector 𝐌 is equal
to the square root of six squared plus 24 squared. This is equal to six root 17. We can repeat this process for
vector 𝐍. The magnitude of vector 𝐍 is the
square root of 16 squared plus negative four squared. This is equal to four root 17. We can, therefore, conclude that
option (B) is also incorrect. The magnitude of vector 𝐌 is not
equal to the magnitude of vector 𝐍.
Option (C) states that vector 𝐌 is
parallel to vector 𝐍. For this to be true, we know that
vector 𝐌 must be called to some scalar 𝑘 multiplied by vector 𝐍. We will now clear some space to see
if this is correct. For vectors 𝐌 and 𝐍 to be
parallel, six 𝐢 plus 24𝐣 must be equal to 𝑘 multiplied by 16𝐢 minus four 𝐣,
where 𝑘 is some nonzero scalar. When multiplying a vector by a
scalar, we simply multiply each of the individual components by that scalar. Therefore, the right-hand side
becomes 16 𝑘𝐢 minus four 𝑘𝐣.
We can now equate the 𝐢- and
𝐣-components. In the 𝐢-components, we have six
is equal to 16𝑘. For the 𝐣-components, 24 is equal
to negative four 𝑘. For the two vectors to be parallel,
the value for 𝑘 that solves both equations must be the same. We can divide both sides of the
first equation by 16 such that 𝑘 is equal to six over 16. Dividing the numerator and
denominator by two, this fraction simplifies to three over eight or
three-eighths. In the second equation, we can
divide both sides by negative four such that 𝑘 is equal to negative 24 over
four. This is equal to negative six. As the value of 𝑘 is different for
both components, vector 𝐌 is not equal to some scalar 𝑘 multiplied by vector
𝐍. We can, therefore, conclude that
the two vectors 𝐌 and 𝐍 are not parallel.
Option (D) says that the vectors 𝐌
and 𝐍 are perpendicular. And we know this is true If their
dot product is equal to zero. To calculate the dot product, we
multiply the 𝐢- and 𝐣-components separately and then find the sum of these
values. The dot product always gives us a
scalar quantity and in this case is equal to six multiplied by 16 plus 24 multiplied
by negative four. Six multiplied by 16 is equal to
96. 24 multiplied by negative four is
negative 96. Adding 96 and negative 96 is the
same as subtracting 96 from 96. This is equal to zero. The dot product of vector 𝐌 and
vector 𝐍 equals zero. This means that these two vectors
are perpendicular and the correct answer is option (D).
If vector 𝐀 is equal to 11𝐢 plus
10𝐣, vector 𝐁 is equal to negative five 𝐢 plus 14𝐣, vector 𝐌 is equal to 𝐀
plus 𝐁, and vector 𝐍 is equal to 𝐀 minus 𝐁, then vector 𝐌 is perpendicular to
vector 𝐍.
