What is the Angle Between Vectors Using the Dot Product Formula?

In summary, to find the angle between the vectors v=-5\sqrt{3}i+5j and w=5i, we can use the dot product formula, where the dot product is a scalar quantity. To calculate the dot product of two vectors given in component form, we would plug in the values of v and w into the formula (a \vec{i} + b \vec{j}) \cdot (c \vec{i} + d \vec{j}) = ac + bd.
  • #1
brinlin
13
0
Find the angle between the vectors \(\displaystyle v=-5\sqrt{3}i+5j\) and \(\displaystyle w=5i\)
 
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  • #2
I'd use the dot product formula ...

$\cos{\theta} = \dfrac{\vec{v} \cdot \vec{w}}{|v| \, |w|}$
 
  • #3
when we use the dot product formula. What would we plug in for v and w.
 
  • #4
to calculate the dot product of two vectors given in component form …

$(a \vec{i} + b \vec{j}) \cdot (c \vec{i} + d \vec{j}) = ac + bd$

… note the dot product is a scalar quantity
 
  • #5
brinlin said:
when we use the dot product formula. What would we plug in for v and w.
? YOU said, in your first post that
$v= -5\sqrt{3}i+ 5j$
$w= 5i$.
 

1. What is the formula for finding the angle between two vectors?

The formula for finding the angle between two vectors is given by the dot product of the two vectors divided by the product of their magnitudes, and then taking the inverse cosine of that value. This can be represented as: θ = cos-1((a · b) / (|a| * |b|))

2. How do you find the angle between two vectors in three-dimensional space?

To find the angle between two vectors in three-dimensional space, you can use the same formula as in two-dimensional space. However, you will need to use the dot product of the two vectors in their three-dimensional form. This can be represented as: θ = cos-1((ax * bx + ay * by + az * bz) / (|a| * |b|))

3. Can the angle between two vectors be negative?

No, the angle between two vectors cannot be negative. The angle is always measured in a counterclockwise direction and will always be between 0 and 180 degrees.

4. What is the difference between the angle between two vectors and the angle of rotation?

The angle between two vectors is the measure of the angle between the two vectors themselves, while the angle of rotation is the measure of the angle needed to rotate one vector onto the other. The angle of rotation can be calculated by subtracting the angle between the two vectors from 360 degrees.

5. Can the angle between two vectors be greater than 180 degrees?

No, the angle between two vectors cannot be greater than 180 degrees. This is because the dot product of two vectors will always be less than or equal to the product of their magnitudes, resulting in an inverse cosine value that is less than or equal to 1. Therefore, the angle between two vectors will always be between 0 and 180 degrees.

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