Find the angles of the triangle formed by vectors #vecP=5hati3hatj+hatk#, #vecQ=2hati+hatj+5hatk# and #vecR=9hati+5hatj+0hatk#?
The triangle formed is a right isoceles triangle and
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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:

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} ]

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} ]

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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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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