Problem 35
Question
Use vectors to find the interior angles of the triangle with the given vertices. $$(1,2), (3,4), (2,5)$$
Step-by-Step Solution
Verified Answer
To conclude, the interior angles of the triangle with the given vertices can be calculated by forming vectors along the sides of the triangle, then calculating their dot products and magnitudes and applying the formula of cosine of angle between two vectors.
1Step 1: Calculate the vectors
First, form three vectors using given points, which will represent the three sides of the triangle. For instance, \(\vec{AB}=(x_2-x_1, y_2-y_1) = (3-1, 4-2) = (2,2)\), and similarly calculate other vectors; \(\vec{BC}\) and \(\vec{CA}\).
2Step 2: Calculate dot products and magnitudes
Calculate the dot product of each pair of vectors using the formula: \(\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y\). And also compute the magnitude of each vector using the formula: \(|\vec{A}| = \sqrt{(A_x)^2 + (A_y)^2}\).
3Step 3: Use the cosine formula
Apply the formula of cosine of the angle between two vectors: \(cos(\Theta) = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}|*|\vec{B}|}\), then take the arc cosine to find the angle in degrees for each pair of vectors.
4Step 4: Calculate the angles of triangle
Iterate the same process for all other pairs of vectors, i.e. BC and CA, and AB and AC, to get the other two interior angles of the triangle, thus completing the solution.
Key Concepts
Interior Angles of a TriangleDot ProductMagnitude of a VectorCosine of an Angle
Interior Angles of a Triangle
Triangles have three interior angles, and their sum is always 180 degrees. In vector mathematics, we can calculate these angles using vector dot products and magnitudes.
The vertices of the triangle form vector representations of the sides. For example, given vertices \((1,2), (3,4), (2,5)\), we can create vectors such as \( \vec{AB} = (3-1, 4-2) = (2,2) \).
After forming vectors for each side, we calculate the angles between them to find the interior angles of the triangle. This process helps in understanding the structure and relationships between various vectors.
The vertices of the triangle form vector representations of the sides. For example, given vertices \((1,2), (3,4), (2,5)\), we can create vectors such as \( \vec{AB} = (3-1, 4-2) = (2,2) \).
After forming vectors for each side, we calculate the angles between them to find the interior angles of the triangle. This process helps in understanding the structure and relationships between various vectors.
Dot Product
The dot product is a way to multiply two vectors, resulting in a scalar value. This scalar can give insights into the relationship between the two vectors.
The formula of the dot product is: \(\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y\).
It measures how much one vector goes in the direction of another. If the dot product is zero, the vectors are perpendicular.
The dot product also plays a crucial role in calculating angles between vectors as it relates directly to the angle's cosine.
The formula of the dot product is: \(\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y\).
It measures how much one vector goes in the direction of another. If the dot product is zero, the vectors are perpendicular.
The dot product also plays a crucial role in calculating angles between vectors as it relates directly to the angle's cosine.
Magnitude of a Vector
The magnitude of a vector tells us its length or size. It's crucial for understanding the vector's properties.
To calculate the magnitude of a vector \( \vec{A} = (A_x, A_y) \), use the formula: \(|\vec{A}| = \sqrt{(A_x)^2 + (A_y)^2}\).
This formula is derived from the Pythagorean theorem, as a vector's magnitude is like the hypotenuse of a right triangle.
To calculate the magnitude of a vector \( \vec{A} = (A_x, A_y) \), use the formula: \(|\vec{A}| = \sqrt{(A_x)^2 + (A_y)^2}\).
This formula is derived from the Pythagorean theorem, as a vector's magnitude is like the hypotenuse of a right triangle.
- The magnitude helps in normalizing vectors, which scales them to have a length of one.
- It provides the denominator in the cosine formula for determining the angle between vectors.
Cosine of an Angle
Cosine is one of the primary trigonometric functions that relates the angle of a triangle to the lengths of its sides.
In vector terms, the cosine of the angle between two vectors can be determined using the dot product.
The formula is: \(\cos(\Theta) = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| \cdot |\vec{B}|}\).
This helps us find the angle between vectors, revealing how aligned they are with each other.
In vector terms, the cosine of the angle between two vectors can be determined using the dot product.
The formula is: \(\cos(\Theta) = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| \cdot |\vec{B}|}\).
This helps us find the angle between vectors, revealing how aligned they are with each other.
- A cosine value near 1 means vectors are pointing in nearly the same direction.
- A value near -1 means they point in opposite directions.
Other exercises in this chapter
Problem 34
Determine whether the Law of sines or the Law of cosines can be used to find another measure of the triangle. Then solve the triangle. $$a=10, \quad b=12, \quad
View solution Problem 34
Represent the complex number graphically, and find the trigonometric form of the number. $$-\frac{5}{2}(\sqrt{3}+i)$$
View solution Problem 35
Determine whether the Law of sines or the Law of cosines can be used to find another measure of the triangle. Then solve the triangle. $$A=24^{\circ}, \quad a=4
View solution Problem 35
Represent the complex number graphically, and find the trigonometric form of the number. $$-7+4 i$$
View solution