Question

Two unit vectors are parallel. What can you deduce about their scalar product?

Short answer

Short answer: The scalar product of two parallel unit vectors A and B can be either 1 or -1. This depends on whether they point in the same direction (scalar product is 1) or opposite directions (scalar product is -1).

Step-by-step solution

Recall the definition of scalar product

Scalar product (also called dot product) is a binary operation that takes two vectors and returns a scalar. For any two vectors A and B, the scalar product A·B can be defined as: A·B = |A||B|cos(θ) Where |A| and |B| are the magnitudes (lengths) of vectors A and B, and θ is the angle between them.

Remember properties of unit vectors

A unit vector is a vector with magnitude (length) equal to 1. If a vector is a unit vector, this means |A| = 1 or |B| = 1.

Know properties of parallel vectors

If two vectors are parallel, this means they point in the same direction or opposite directions. Therefore, the angle between them θ should be either 0 degrees (if pointing in the same direction) or 180 degrees (if pointing in opposite directions). In radians, we have θ = 0 or θ = π.

Deduce the scalar product of parallel unit vectors

Now we can use the information from steps 1-3 to deduce the scalar product of our given parallel unit vectors A and B: A·B = |A||B|cos(θ) Since A and B are unit vectors, we have: A·B = 1*1*cos(θ) Now, since vectors A and B are parallel, we have θ = 0 or θ = π. If θ = 0, then cos(0) = 1, and we have: A·B = 1 If θ = π, then cos(π) = -1, and we have: A·B = -1 So, the scalar product of two parallel unit vectors A and B can be either 1 or -1, depending on whether they point in the same direction or opposite directions.

Key concepts

Dot Product

When talking about the dot product, also known as the scalar product, we're dealing with a fundamental operation in vector mathematics. It combines two vectors to produce a single number – a scalar. The dot product is given by the equation:
\[\begin{equation}A \cdot B = |A||B|\cos(\theta)d{equation}
Here, |A| and |B| represent the magnitudes of vectors A and B, respectively, while \( \theta \) expresses the angle between them. Another critical aspect of the dot product is its ability to measure how much one vector extends in the direction of another, offering insights into vector alignment and their parallels.

Unit Vectors

Unit vectors hold a special place in the world of vectors due to their simplicity and usefulness. They are vectors that have a magnitude of exactly one. No matter what direction they point in, the length is always one. This special property simplifies many vector operations. For example:
\[\begin{equation} \vec{u} = \frac{\vec{v}}{|\vec{v}|} d{equation}
Constructing a unit vector, or normalizing a vector, involves dividing the vector by its own magnitude, which converts it into a unit vector \( \vec{u} \) that maintains the direction of the original vector \( \vec{v} \).

Parallel Vectors

Vectors that are parallel share a commonality in their direction. They can be parallel in two ways - either pointing in the same direction or diametrically opposed. One intuitive way to understand parallel vectors is by considering their direction vectors. If one vector is a multiple of the other, they're parallel. Specifically, for two vectors to be parallel, the angles between them must be either \( 0 \) degrees (same direction) or \( 180 \) degrees (opposite direction), directly affecting their scalar product as we've learned.

Vector Magnitudes

When discussing vector magnitudes, we're speaking of how we measure the 'length' or 'size' of a vector. The magnitude is a non-negative scalar and represents the distance from the vector's initial point to its terminal point in a multi-dimensional space. Calculating a vector's magnitude in a 2D or 3D space can be done with the Pythagorean theorem:
\[\begin{equation}|A| = \sqrt{a^2 + b^2}d{equation}
where \( a \) and \( b \) are the components along the x and y axes. Understanding magnitudes is crucial as it provides the building blocks for various vector operations, including dot product computations.

Angles Between Vectors

The angle between vectors is a fundamental concept encapsulating the spatial relationship between two directions. It's critical when determining the dot product along with the magnitudes. The cosine of the angle between vectors is precisely what the dot product measures when normalized by the vectors' lengths. This relationship allows us to deduce crucial properties about the vectors' directions: whether they are pointing the same way (\( \theta = 0 \)), opposite ways (\( \theta = 180 \) degrees), or somewhere in between. Having the angle, we can interpret the vectors' relative orientations and thus their dot product.