Sketch the coordinate axes and then include the vectors 𝐮, 𝐯, and 𝐮 ×𝐯 as vectors starting at th

step 1 We have to find u cross 3 which is given as j plus 2k into i. This is given as j cross i plus 2

Sketch the coordinate axes and then include the vectors 𝐮,...

Sketch the coordinate axes and then include the vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{u} \times \mathbf{v}$ as vectors starting at the origin.
$$\mathbf{u}=\mathbf{j}+2 \mathbf{k}, \quad \mathbf{v}=\mathbf{i}$$

Key Concepts

Coordinate Axes and 3D Space

The coordinate axes in a three-dimensional system form the foundation for representing vectors in 3D space. They include the x-, y-, and z-axes, which intersect at the origin and provide a reference frame for describing the direction and magnitude of any vector. Understanding these axes is fundamental to visualizing and sketching vectors, as well as performing vector operations.

Standard Unit Vectors

Standard unit vectors, typically denoted as i, j, and k, represent the unit length in the directions of the x-, y-, and z-axes, respectively. They serve as basic building blocks for expressing more complex vectors as combinations of these directions. Their clear definition simplifies operations like vector addition, scalar multiplication, and the calculation of vector products.

Vector Representation

Vector representation involves illustrating vectors as directed line segments (arrows) that originate from a common starting point, usually the origin. This graphical depiction helps in visualizing the magnitude and direction of the vector, making it easier to understand geometric relations, perform vector addition, and interpret the results of vector operations.

Vector Cross Product

The cross product is a binary operation on two vectors in three-dimensional space that results in a third vector. This resulting vector is orthogonal (perpendicular) to both of the original vectors, and its magnitude is proportional to the sine of the angle between them and the product of their magnitudes. The geometrical and physical significance of the cross product makes it a key concept in vector calculus and applications such as torque and rotational motion.

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