Find the angles of the triangle formed by vectors #vecP=5hati-3hatj+hatk#, #vecQ=-2hati+hatj+5hatk# and #vecR=9hati+5hatj+0hatk#?

Answer 1

The triangle formed is a right isoceles triangle and #m/_A=90^@#, #m/_B=m/_C=45^@#

Let the three vectors #vecP=5hati-3hatj+hatk#, #vecQ=-2hati+hatj+5hatk# and #vecR=9hati+5hatj+0hatk#, form a triangle with vertices #a#, #B# and #C# respectively.
Hence #|AB|=|vecQ-vecP|=|(-2-5)hati+(1-(-3))hatj+(5-1)hatk|#
= #|-7hati+4hatj+4hatk|=sqrt((-7)^2+4^2+4^2)=sqrt81=9#
#|BC|=|vecR-vecQ|=|(9-(-2))hati+(5-1)hatj+(0-5)hatk|#
= #|11hati+4hatj-5hatk|=sqrt(11^2+4^2+(-5)^2)=sqrt162=9sqrt2#
and #|CA|=|vecR-vecP|=|(9-5)hati+(5-(-3))hatj+(0-1)hatk|#
= #|4hati+8hatj-1hatk|=sqrt(4^2+8^2+(-1)^2)=sqrt81=9#
It is apparent that #AB^2+CA^2=BC^2#, where #AB=CA#
Hence the triangle formed is a right isoceles triangle and #m/_A=90^@#, #m/_B=m/_C=45^@#
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Answer 2

To find the angles of the triangle formed by vectors (\vec{P}), (\vec{Q}), and (\vec{R}), you can use the dot product and vector magnitude.

The angle between two vectors (\vec{A}) and (\vec{B}) can be calculated using the formula: [ \cos(\theta) = \frac{\vec{A} \cdot \vec{B}}{\lVert \vec{A} \rVert \lVert \vec{B} \rVert} ]

Where:

  • (\theta) is the angle between the vectors.
  • (\vec{A} \cdot \vec{B}) denotes the dot product of vectors (\vec{A}) and (\vec{B}).
  • (\lVert \vec{A} \rVert) denotes the magnitude of vector (\vec{A}).

Using this formula, you can find the angles between vectors (\vec{P}), (\vec{Q}), and (\vec{R}) respectively:

  1. Angle between vectors (\vec{P}) and (\vec{Q}): [ \cos(\theta_{PQ}) = \frac{\vec{P} \cdot \vec{Q}}{\lVert \vec{P} \rVert \lVert \vec{Q} \rVert} ]

  2. Angle between vectors (\vec{Q}) and (\vec{R}): [ \cos(\theta_{QR}) = \frac{\vec{Q} \cdot \vec{R}}{\lVert \vec{Q} \rVert \lVert \vec{R} \rVert} ]

  3. Angle between vectors (\vec{R}) and (\vec{P}): [ \cos(\theta_{RP}) = \frac{\vec{R} \cdot \vec{P}}{\lVert \vec{R} \rVert \lVert \vec{P} \rVert} ]

Finally, you can find the angles by taking the inverse cosine of the values obtained: [ \text{Angle} = \cos^{-1}(\text{value}) ]

Compute these values and then find the corresponding angles.

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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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