Find the angles between the vectors in Exerciscs 9-12 to the...

Find the angles between the vectors in Exerciscs $9-12$ to the nearest hundredth of a radian.
$$\mathbf{u}=\mathbf{i}+\sqrt{2} \mathbf{j}-\sqrt{2} \mathbf{k}, \quad \mathbf{v}=-\mathbf{i}+\mathbf{j}+\mathbf{k}$$

Key Concepts

Magnitude of a Vector

The magnitude of a vector, often referred to as its length, is calculated using the square root of the sum of the squares of its components. It quantifies the size of the vector and is essential in normalizing vectors or in computing angles between them.

Angle Calculation Using the Dot Product

The angle between two vectors can be computed by relating their dot product to their magnitudes using the formula cos(?) = (u · v) / (||u|| ||v||). This formula allows one to solve for the angle ? by taking the arccosine of the resulting quotient, which provides a measure of the angle in a consistent mathematical framework.

Vector Algebra

Vector algebra is a branch of mathematics that deals with quantities having both magnitude and direction. It involves operations such as addition, scalar multiplication, and products that help in analyzing spatial relationships between vectors.

Dot Product

The dot product is an algebraic operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components of the vectors and then summing these products. The dot product is fundamental in determining the angle between two vectors because it relates directly to their magnitudes and the cosine of the angle between them.

Radian Measure

Radians are a unit of angular measurement based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius. This unit is frequently used in calculus and physics due to its natural relationship with various mathematical functions, and it is preferred in many scientific computations, including those involving angles between vectors.

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