Question

Two vectors have modulus 10 and 12 . The angle between them is $\frac{\pi }{3}$. Find their scalar product.

Short answer

Answer: The scalar product of the two vectors is 60.

Step-by-step solution

Write the scalar product formula for magnitudes and the angle

The scalar product( also called dot product) of two vectors can be found using magnitudes and the angle between them. The formula is: $$\mathbf{\text{A}}\cdot \mathbf{\text{B}}=\Vert \mathbf{\text{A}}\Vert \Vert \mathbf{\text{B}}\Vert \cos \theta$$ where $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$ are the two vectors, $\Vert \mathbf{\text{A}}\Vert$ and $\Vert \mathbf{\text{B}}\Vert$ are their magnitudes, and $\theta$ is the angle between the two vectors.

Plug in the given values

We are given that $\Vert \mathbf{\text{A}}\Vert =10$ and $\Vert \mathbf{\text{B}}\Vert =12$. The angle between them is $\theta =\frac{\pi }{3}$. Substituting these values into the formula, we get: $$\mathbf{\text{A}}\cdot \mathbf{\text{B}}=10\times 12\times \cos \frac{\pi }{3}$$

Calculate the scalar product

The cosine of $\frac{\pi }{3}$ is $\frac{1}{2}$. So, our scalar product is: $$\mathbf{\text{A}}\cdot \mathbf{\text{B}}=10\times 12\times \frac{1}{2}$$ Multiplying, we get: $$\mathbf{\text{A}}\cdot \mathbf{\text{B}}=60$$ In conclusion, the scalar product of the two vectors is 60.

Key concepts

Vectors

Vectors are mathematical entities that have both magnitude and direction. They are often used in physics and engineering to represent quantities like force, velocity, and displacement. Imagine a vector as an arrow: the length of the arrow represents the magnitude, while the direction in which the arrow points indicates the direction of the vector.

Key characteristics of vectors include:

• Magnitude: The size or length of the vector.
• Direction: The orientation in space along which the vector acts.
• Representation: Vectors can be represented graphically as arrows or algebraically as coordinate pairs or triplets, such as $\mathbf{\text{A}}=(a_1,a_2)$ for 2D vectors or $\mathbf{\text{B}}=(b_1,b_2,b_3)$ for 3D vectors.

Understanding vectors is essential in solving problems involving quantities that have direction, such as in physics where forces are resolved into vector components. They form the foundational concepts for further exploration into dot products and angles between vectors.

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors to result in a scalar. This means the outcome has magnitude but no direction. This product gives useful insight into the relationship between two vectors, especially in terms of their angle.

The formula for the dot product of two vectors $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$ is:
$$\mathbf{\text{A}}\cdot \mathbf{\text{B}}=\Vert \mathbf{\text{A}}\Vert \Vert \mathbf{\text{B}}\Vert \cos \theta$$
Here:

• $\Vert \mathbf{\text{A}}\Vert$: Represents the magnitude of vector $\mathbf{\text{A}}$.
• $\Vert \mathbf{\text{B}}\Vert$: Represents the magnitude of vector $\mathbf{\text{B}}$.
• $\theta$: The angle between the vectors $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$.

Using the dot product can tell us how parallel or perpendicular vectors are to each other. For instance:

• If the dot product is zero, the vectors are perpendicular.
• If the dot product is positive, they have a component facing the same general direction.
• If negative, they point in opposite directions.

Hence, the dot product is integral not just for calculating angles, but also for understanding the geometric relationship between vectors.

Angle Between Vectors

The angle between vectors is a measure of how "tilted" they are relative to each other. Knowing this angle can help in many applications like physics or computer graphics, where orientation plays a critical role.

This angle, represented by $\theta$, is part of the dot product formula:
$$\mathbf{\text{A}}\cdot \mathbf{\text{B}}=\Vert \mathbf{\text{A}}\Vert \Vert \mathbf{\text{B}}\Vert \cos \theta$$
From this equation, you can also solve for $\theta$:
$$\cos \theta =\frac{\mathbf{\text{A}}\cdot \mathbf{\text{B}}}{\Vert \mathbf{\text{A}}\Vert \Vert \mathbf{\text{B}}\Vert }$$

Characteristics of the angle include:

• It ranges from $0$ to $\pi$ radians, where $0$ represents parallel vectors and $\pi$ represents opposite direction vectors.
• If two vectors are perpendicular, $\theta$ is $\frac{\pi }{2}$, making the dot product zero.

Understanding these angles is crucial when determining the relationship between two vectors. Knowing whether they are parallel, at an angle, or perpendicular helps in various fields such as navigation, robotics, and 3D rendering.