MAT188H1 Chap.2 Norms, Dot Products and Angles
Norms, Dot Products and Angles
Length comes from a vector paired with itself
The norm of a real vector is the square root of the sum of its squared components. It extends the familiar length of a planar arrow to vectors with any number of components. Squaring matters: negative coordinates contribute positively to length. The norm is a scalar, not a new vector, and the final square root is essential.
A sum of squares describes squared length until that last operation is performed.
The zero vector has norm zero, and a nonzero vector has positive norm. This distinction determines when division by a norm is allowed. Before calculating an angle, check whether either input is zero. An expression with a zero denominator does not become a valid angle by inserting a convenient value.
Geometric language about direction must follow the conditions of the formula being used.
The dot product combines matching coordinates
For vectors of the same dimension, the dot product multiplies corresponding entries and adds the resulting products. Unlike a norm calculation, the products may have either sign. Cancellation is meaningful: it can indicate orthogonality rather than an error in the arithmetic.
The result is again a scalar. Keep this output type separate from vector addition, which returns a vector with the original number of components.
Taking the dot product of a vector with itself gives its squared norm. This identity offers an independent check on both calculations. If your self dot product is negative, an arithmetic or transcription error has occurred.
If you find the norm by a square root, keep the positive root because length is nonnegative. A negative coordinate does not require a negative choice of square root.
Separate the numerator from the denominator
For nonzero vectors, the cosine of the angle is their dot product divided by the product of their norms. Compute those quantities separately before combining them.
The numerator records signed alignment, while the positive denominator removes the effect of length. Only after simplifying the ratio should you apply inverse cosine. The angle lies between zero and a straight angle, consistent with the ordinary angle between arrows placed at a common tail.
A frequent error replaces the product of norms with their sum, or substitutes squared norms without square roots.
Either change produces a different expression. Exact radicals are useful because they make cancellation visible and avoid premature rounding. A cosine ratio outside the interval from negative one to one indicates that the calculation needs checking; it is not an instruction to invent an angle.
Orthogonality is an algebraic test
Two vectors are orthogonal when their dot product is zero.
For nonzero vectors this corresponds to a right angle. The algebraic definition also covers the zero vector, whose dot product with every vector is zero. Its angle with another vector is still undefined because the angle formula would divide by zero.
Keeping these conclusions separate resolves an apparent contradiction between the definition and the geometric picture.
Positive and negative dot products also provide a fast interpretation before a calculator is used. With nonzero inputs, the denominator in the cosine formula is positive, so the numerator controls the cosine's sign.
A positive result indicates an acute angle, and a negative result indicates an obtuse angle. If a numerical inverse cosine produces the opposite classification, check the calculator mode and the signs in the numerator.
Parallelism and perpendicularity ask different questions
A scalar multiple test compares complete vectors, while an orthogonality test compares their dot product with zero.
Equal lengths alone settle neither relationship. Two arrows can have the same length and different directions. Conversely, parallel vectors may have different lengths. Choose the test from the requested relationship rather than from whichever formula you remember first.
For a clear solution, retain the exact dot product and both norms in the written work.
Then give the cosine ratio, the angle when defined, and a short geometric interpretation. This sequence lets another reader locate the first incorrect operation instead of having to reconstruct the calculation from a final decimal. It also makes the exceptional zero input visible before any division occurs.
What this chapter covers
- 01
Norm and squared norm
- 02
Dot product and orthogonality
- 03
Angle formula and zero-vector exception
Worked example · free
An obtuse angle from an exact ratio
- 2The squared norms are 9 and 8. Their nonnegative square roots give ||u|| = 3 and ||v|| = 2√2. Both inputs are nonzero, so the angle formula is defined.
- 2Multiply matching entries and add: u·v = 3(-2) + 0(2) + 0(0) = -6. The negative sign predicts an obtuse angle before evaluating inverse cosine.
- 2Divide by the product of lengths: cos θ = -6/(3 × 2√2) = -1/√2. Thus θ = 3π/4 radians, or 135 degrees. This agrees with the sign prediction.
Key terms
- Norm
- The nonnegative length assigned to a vector by the square root of the sum of squared components. It measures magnitude while discarding the signs that specify coordinate direction.
- Dot Product
- The scalar obtained by adding products of corresponding vector components. Its value connects component arithmetic to angles and supplies the test for whether two vectors are orthogonal.
- Orthogonal Vectors
- Vectors whose dot product equals zero. Nonzero orthogonal vectors meet at a right angle; the zero vector satisfies the algebraic condition even though an angle involving it is undefined.
- Vector Angle
- The angle defined for nonzero vectors by inverse cosine of the normalized dot product. The normalization divides by both lengths, so a change in magnitude alone does not change direction.
- Squared Norm
- The dot product of a vector with itself, equal to the sum of its squared components. It is useful for arithmetic checks but must be square rooted before it represents length.
Norms, Dot Products and Angles FAQ
Why can perpendicular vectors have negative components?
Orthogonality depends on the sum of component products, not on every component being positive. Positive and negative contributions can cancel exactly. A sign pattern by itself cannot decide the relationship; carry out the complete dot product, keeping each signed multiplication visible until the final sum.
Must I calculate an angle to prove orthogonality?
A zero dot product already establishes orthogonality by definition. Inverse cosine adds unnecessary work for that question and would fail for a zero input. If the question separately requests the geometric angle, first establish that both vectors are nonzero, then interpret the orthogonality as a right angle.
How should I handle a cosine slightly larger than one?
Revisit the exact numerator and denominator before trusting a rounded value. Check matching coordinates, square roots, and multiplication of the two norms. Early decimal rounding can obscure an exact endpoint such as one. Retaining radicals until the final step helps distinguish rounding effects from an incorrect formula.
Exam move
Build a revision table with inputs, dot product, squared norms and geometric classification. Include a cancelling dot product, an obtuse angle and the zero vector. Explain aloud why each denominator is allowed or forbidden. Recompute one norm through the self dot product rather than repeating exactly the same written calculation.
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