Maths with Lemon

Vectors- Basic principles

"VECTOR!! That's me. 'Cause I'm committing crimes with both direction and magnitude. OH YEAH!!!!!!!!!!!" - As he introduces himself to Gru in the bank. "

Despicable Me (Movie)

Adding and subtracting vectors

What you have to know:

  • Addition and subtraction of numbers.

Key Points

  • 1. Watch the video:

  • 2. Vectors in 3 dimensions. Watch this video.

Finding vectors from given ones

What you have to know:

  • The Pythagorean Theorem.

Key Points

  • 1. Watch the video:

  • 2. Solved example with pathways and proofs with vectors. Watch the video.

Magnitude of a vector

What you have to know:

  • The Pythagorean Theorem.

Key Points

  • 1. Watch the video:

  • 2. Magnitude of a 3D vector. Watch this video.

Position Vectors

What you have to know:

  • The Pythagorean Theorem.

Key Points

  • 1. Watch the video:

Direction of a vector

What you have to know:

  • Trigonometric calculations of \(\sin \theta\), \(\cos \theta\) and \(\tan \theta\).

Key Points

  • 1. Watch the video:

Unit vectors i and j

What you have to know:

  • Trigonometric calculations of \(\sin \theta\), \(\cos \theta\) and \(\tan \theta\).

Key Points

  • 1. Watch the video:

  • 2. Unit vectors example solved. Watch this video.

Extra

Material and references:

  • Hodder Book HL(ISBN: 9781510462366) :
    8A, 8B, 8C

Key Points

  • 1. You should be able to represent vectors as either directed line segments or by their components (as column vectors or using \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) base vectors), and add, subtract, and multiply vectors by a scalar using both representations.

  • 2. The magnitude of a vector \[ \mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \] is \[ |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}, \] and the unit vector in the same direction as \( \mathbf{a} \) is \( \dfrac{\mathbf{a}}{|\mathbf{a}|} \).

  • 3. The position vector of a point \( A \) is \( \mathbf{a} = \overrightarrow{OA} \). The displacement vector from \( A \) to \( B \) is \( \overrightarrow{AB} = \mathbf{b} - \mathbf{a} \), and the distance between \( A \) and \( B \) is \( |\mathbf{b} - \mathbf{a}| \).

  • 4. If points \( A \) and \( B \) have position vectors \( \mathbf{a} \) and \( \mathbf{b} \), then the midpoint of \( AB \) has position vector \( \tfrac{1}{2}(\mathbf{a} + \mathbf{b}) \).

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