Question
Do not round any Intermediate computations Round your answers to the nearest degree. Complete the following for each pair of vectors. 1.Let w=-3i+2j and r=-6i+4j (B) Find the angle between w and r. square ^circ (b) Determine whether w and r are parallel, orthogonal, or neither. parallel orthogonal neither 2. Let t=-5i-4j and v=-6i+5j (a) Find the angle between t and v. square ^circ (b) Determine whether t and V are parallel, orthogonal, or neither. parallel orthogonal neither
Solution
4
(44 Votos)
Miguel
Mestre · Tutor por 5 anos
Resposta
1. (B) To find the angle between vectors
and
, we can use the dot product formula:
where
is the angle between the vectors,
is the dot product of
and
, and
and
are the magnitudes of
and
respectively.First, let's calculate the dot product of
and
:
Next, let's calculate the magnitudes of
and
:
Now, we can substitute these values into the formula for \( \cos(\theta) \):
Since \( \cos(\theta) = 1 \), the angle
is 0 degrees.(B) To determine whether
and
are parallel, orthogonal, or neither, we can compare their direction ratios. If the direction ratios are proportional, then the vectors are parallel. If the direction ratios are perpendicular, then the vectors are orthogonal. Otherwise, they are neither.In this case, the direction ratios of
and
are proportional, as
is simply twice
. Therefore,
and
are parallel.2. (A) To find the angle between vectors
and
, we can use the same dot product formula as in the previous question:
First, let's calculate the dot product of
and
:
Next, let's calculate the magnitudes of
and
:
Now, we can substitute these values into the formula for \( \cos(\theta) \):
To find the angle
, we take the inverse cosine of \( \cos(\theta) \):
Using a calculator, we find that
.(A) To determine whether
and
are parallel, orthogonal, or neither, we can again compare their direction ratios. In this case, the direction ratios of
and
are not proportional, and they are not perpendicular either. Therefore,
and
are neither parallel nor orthogonal.