Learn & Review: Lec 1 | MIT 18.02 Multivariable Calculus

Summary of Vector Introduction and Dot Product

This content provides an introduction to vectors, their properties, and the dot product operation, primarily aimed at students in an introductory multivariable calculus course.

1. Introduction to Vectors

• Definition: A vector is a quantity possessing both direction and magnitude (or length).
• Representation:
  • Geometrically, vectors are drawn as arrows.
  • Numerically, vectors are represented by their components along coordinate axes (e.g., x, y, z).
• Notation:
  • Vectors are often denoted with an arrow on top (e.g., $\vec{a}$) or in bold in textbooks.
  • Unit vectors along the coordinate axes are denoted by $\hat{i}$ (x-axis), $\hat{j}$ (y-axis), and $\hat{k}$ (z-axis).
  • A vector $\vec{a}$ can be expressed in component form as $a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ or using bracket notation, e.g., $[a_1, a_2, a_3]$ or $\langle a_1, a_2, a_3 \rangle$.
• Scalar Quantity: A quantity with magnitude only, as opposed to a vector quantity.
• Vector Magnitude (Length): Denoted by $|\vec{a}|$. For a vector $\vec{a} = \langle a_1, a_2, a_3 \rangle$, the length is calculated using the Pythagorean theorem: $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
  • Example: For $\vec{a} = \langle 3, 2, 1 \rangle$, the length is $|\vec{a}| = \sqrt{3^2 + 2^2 + 1^2} = \sqrt{9 + 4 + 1} = \sqrt{14}$.
• Vector Direction: Can be obtained by scaling the vector to unit length (dividing by its magnitude).
• Vector Translation: A vector does not have a fixed starting point; it can be translated anywhere in space while maintaining its direction and magnitude.

2. Vector Operations

• Scalar Multiplication: Multiplying a vector by a scalar changes its magnitude (and potentially direction if the scalar is negative).
  • Example: $2\vec{a}$ is a vector twice as long as $\vec{a}$ in the same direction. $-\vec{a}$ is a vector of the same length as $\vec{a}$ but in the opposite direction.
• Vector Addition:
  • Geometrically: To add vectors $\vec{a}$ and $\vec{b}$, place the tail of $\vec{b}$ at the head of $\vec{a}$. The resultant vector $\vec{a} + \vec{b}$ goes from the tail of $\vec{a}$ to the head of $\vec{b}$. This forms a parallelogram, and $\vec{a} + \vec{b}$ is the diagonal.
  • Numerically: Add the corresponding components: If $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$, then $\vec{a} + \vec{b} = \langle a_1+b_1, a_2+b_2, a_3+b_3 \rangle$.
  • Commutativity: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$.
• Vector Subtraction: $\vec{a} - \vec{b}$ is equivalent to adding the opposite vector: $\vec{a} + (-\vec{b})$.
  • Geometrically, if you draw $\vec{a}$ and $\vec{b}$ from the same origin, $\vec{a} - \vec{b}$ is the vector from the head of $\vec{b}$ to the head of $\vec{a}$.

3. The Dot Product

• Definition: The dot product (or scalar product) of two vectors $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$ is a scalar quantity defined as: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$
  • Key Point: The result is a scalar (a number), not a vector.
• Geometric Interpretation: The dot product is also equal to the product of the magnitudes of the two vectors and the cosine of the angle $\theta$ between them: $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$
• Relationship between Definitions: The equality of these two definitions can be proven using the Law of Cosines.
  • $\vec{a} \cdot \vec{a} = |\vec{a}|^2$ (since $\cos 0 = 1$).
  • The Law of Cosines applied to a triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a}-\vec{b}$ leads to the dot product formula.
• Applications of the Dot Product:
  • Calculating Lengths and Angles: The formula $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$ allows us to find the angle between two vectors: $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}$
    • Example: For points P(1,0,0), Q(0,1,0), R(0,0,2), to find the angle at P in triangle PQR:
      • $\vec{PQ} = \langle -1, 1, 0 \rangle$, $|\vec{PQ}| = \sqrt{2}$
      • $\vec{PR} = \langle -1, 0, 2 \rangle$, $|\vec{PR}| = \sqrt{5}$
      • $\vec{PQ} \cdot \vec{PR} = (-1)(-1) + (1)(0) + (0)(2) = 1$
      • $\cos \theta = \frac{1}{\sqrt{2}\sqrt{5}} = \frac{1}{\sqrt{10}}$
      • $\theta = \arccos\left(\frac{1}{\sqrt{10}}\right) \approx 71.5^\circ$
  • Detecting Orthogonality (Perpendicularity):
    • Two vectors $\vec{a}$ and $\vec{b}$ are perpendicular if and only if their dot product is zero ($\vec{a} \cdot \vec{b} = 0$). This is because $\cos 90^\circ = 0$.
    • Application to Planes: The equation $x + 2y + 3z = 0$ defines a plane passing through the origin. The vector $\vec{n} = \langle 1, 2, 3 \rangle$ is normal (perpendicular) to this plane. Any vector $\vec{OP} = \langle x, y, z \rangle$ lying in the plane will be perpendicular to $\vec{n}$, hence $\vec{OP} \cdot \vec{n} = 0$.
  • Measuring Alignment: The sign of the dot product indicates the general direction of alignment between two vectors:
    • Positive dot product: Vectors point in generally the same direction (acute angle).
    • Zero dot product: Vectors are perpendicular (right angle).
    • Negative dot product: Vectors point in generally opposite directions (obtuse angle).
• Future Application: The dot product can also be used to find the component of a vector along any given direction.