How do you find the angle between the vectors #u=<3, 2># and #v=<4, 0>#?

Answer 1

The angle is # 33.7 °#(3sf)

The angle #theta# between two vectors #vec A# and #vec B# is related to the modulus (or magnitude) and scaler (or dot) product of #vec A# and #vec B# by the relationship:

# vec A * vec B = |A| |B| cos theta #

By convention when we refer to the angle between vectors we choose the acute angle.

So for this problem, let the angle betwen #vecu# and #vecv# be #theta# then:

#vec u=<<3, 2>># and #vec v=<<4, 0>>#

The modulus is given by;

# |vec u| = |<<3, 2>>| = sqrt(3^2+2^2)=sqrt(9+4)=sqrt(13) #
# |vec v| = |<<4, 0>>| = sqrt(4^2+0^2)=sqrt(16)=4 #

And the scaler product is:

# vec u * vec v = <<3, 2>> * <<4, 0>>#
# \ \ \ \ \ \ \ \ \ \ = (3)(4) + (2)(0)#
# \ \ \ \ \ \ \ \ \ \ = (12)#

And so using # vec A * vec B = |A| |B| cos theta # we have:

# 12 = sqrt(13) * 4 * cos theta #
# :. cos theta = (12)/(4sqrt(13))#
# :. cos theta = 0.83205 ... #
# :. theta = 33.69006 °#
# :. theta = 33.7 °#(3sf)

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Answer 2

To find the angle between two vectors ( \mathbf{u} ) and ( \mathbf{v} ), you can use the dot product formula:

[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta) ]

Where:

  • ( \mathbf{u} ) and ( \mathbf{v} ) are the given vectors.
  • ( |\mathbf{u}| ) and ( |\mathbf{v}| ) are the magnitudes of vectors ( \mathbf{u} ) and ( \mathbf{v} ) respectively.
  • ( \theta ) is the angle between the vectors.

First, calculate the dot product of ( \mathbf{u} ) and ( \mathbf{v} ) using the given components:

[ \mathbf{u} \cdot \mathbf{v} = (3)(4) + (2)(0) = 12 ]

Next, calculate the magnitudes of vectors ( \mathbf{u} ) and ( \mathbf{v} ):

[ |\mathbf{u}| = \sqrt{3^2 + 2^2} = \sqrt{13} ] [ |\mathbf{v}| = \sqrt{4^2 + 0^2} = 4 ]

Now, substitute these values into the dot product formula:

[ 12 = (\sqrt{13})(4) \cos(\theta) ]

Solve for ( \theta ) using algebra:

[ \cos(\theta) = \frac{12}{4\sqrt{13}} = \frac{3}{\sqrt{13}} ]

[ \theta = \arccos\left(\frac{3}{\sqrt{13}}\right) ]

This gives the angle between the vectors ( \mathbf{u} ) and ( \mathbf{v} ). You can use a calculator to find its approximate value.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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