The i-j-k notation, also known as the standard basis or unit vector notation, is a way of representing and working with vectors in a three-dimensional Cartesian coordinate system. It provides a standardized and intuitive method for describing the direction and magnitude of vectors using three perpendicular unit vectors: i, j, and k.
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The i-j-k notation represents the three unit vectors in a three-dimensional Cartesian coordinate system: i (1, 0, 0), j (0, 1, 0), and k (0, 0, 1).
Any vector in the three-dimensional space can be expressed as a linear combination of the i, j, and k unit vectors.
The i-j-k notation allows for the easy manipulation and transformation of vectors, such as addition, subtraction, and scalar multiplication.
Vector operations, like the dot product and cross product, can be performed efficiently using the i-j-k notation.
The i-j-k notation is particularly useful in physics and engineering applications, where vectors are commonly used to describe physical quantities like displacement, velocity, and force.
Review Questions
Explain how the i-j-k notation is used to represent vectors in a three-dimensional Cartesian coordinate system.
The i-j-k notation represents the three unit vectors in a three-dimensional Cartesian coordinate system: i (1, 0, 0), j (0, 1, 0), and k (0, 0, 1). Any vector in this three-dimensional space can be expressed as a linear combination of these three unit vectors. For example, a vector $\vec{v} = 2i + 3j - 4k$ represents a vector with a magnitude of $\sqrt{4 + 9 + 16} = 7$ and a direction defined by the coefficients of i, j, and k.
Describe how the i-j-k notation can be used to perform vector operations, such as addition and scalar multiplication.
The i-j-k notation allows for the easy manipulation and transformation of vectors. For vector addition, the corresponding coefficients of i, j, and k are added together. For example, $\vec{u} = 2i + 3j - 4k$ and $\vec{v} = -i + 5j + 2k$ can be added to get $\vec{u} + \vec{v} = (2 - 1)i + (3 + 5)j + (-4 + 2)k = i + 8j - 2k$. Similarly, scalar multiplication can be performed by multiplying the coefficients of i, j, and k by the scalar value. This makes the i-j-k notation a powerful tool for working with vectors in a three-dimensional space.
Explain the importance of the i-j-k notation in physics and engineering applications, and how it relates to the understanding of physical quantities represented by vectors.
The i-j-k notation is particularly useful in physics and engineering applications, where vectors are commonly used to describe physical quantities like displacement, velocity, and force. By representing these vectors in the i-j-k notation, it becomes easier to perform vector operations and transformations, which are essential for analyzing and solving problems in these fields. For example, the i-j-k notation allows for the efficient calculation of the dot product and cross product of vectors, which are fundamental operations used to determine the work done by a force, the torque acting on an object, and other important physical relationships. The intuitive nature of the i-j-k notation makes it a valuable tool for understanding and manipulating vectors in the context of various physics and engineering applications.
A mathematical quantity that has both magnitude (size) and direction, typically represented by an arrow.
Cartesian Coordinate System: A two-dimensional or three-dimensional coordinate system that uses perpendicular axes (x, y, and z) to locate points in space.