Cosine Similarity

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Cosine Similarity #

Linear AlgebraDifficulty: ★★★☆☆Depth: 0Unlocks: 3

A measure of similarity between two vectors defined as the cosine of the angle between them (dot product normalized by norms); used as an attention scoring function and for comparing embeddings. It highlights direction-based similarity independent of vector magnitude.

Interactive Visualization #

⏮◀◀▶▶STEP0.25x1xZOOM

t=0s

Core Concepts #

Key Symbols & Notation #

dot(a,b) (dot product of vectors a and b)||a|| (Euclidean norm/magnitude of vector a)

Essential Relationships #

Unlocks (2) #

Attention Mechanismslvl 5Vector Embeddingslvl 4

Advanced Learning Details

Graph Position #

6

Depth Cost

3

Fan-Out (ROI)

1

Bottleneck Score

0

Chain Length

Cognitive Load #

6

Atomic Elements

25

Total Elements

L0

Percentile Level

L4

Atomic Level

All Concepts (10) #

Teaching Strategy #

Deep-dive lesson - accessible entry point but dense material. Use worked examples and spaced repetition.

When you compare two vectors, you often care less about how big they are and more about whether they “point” in the same direction. Cosine similarity is the standard tool for measuring that directional agreement—and it shows up everywhere from search and embeddings to attention scores in transformers.

TL;DR:

Cosine similarity between two nonzero vectors a and b is

cosSim(a,b)=a⋅b∥a∥ ∥b∥\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}cosSim(a,b)=∥a∥∥b∥a⋅b​

It equals the cosine of the angle between them: 1 means same direction, 0 means orthogonal (no directional alignment), −1 means opposite direction. It’s magnitude-invariant (scaling a vector doesn’t change it), which makes it ideal for comparing embeddings and as an attention scoring function. Be careful: both vectors must be nonzero, and “cosine distance = 1 − cosSim” is commonly used but is not a true metric in general (triangle inequality can fail).

Prerequisites (quick but explicit) #

This node is meant to be foundational, but cosine similarity does assume a few micro-skills. Here’s a compact checklist.

You should recognize these ideas #

ConceptMeaningFormula / Note
Vector aOrdered list of numbersa = (a₁, a₂, …, aₙ)
Dot productMeasures alignment via componentwise multiplication$a⋅b=∑i=1naibi\mathbf{a}\cdot\mathbf{b}=\sum_{i=1}^n a_i b_ia⋅b=∑i=1n​ai​bi​$
Euclidean normVector “length” (magnitude)$$\
Angle interpretationIn geometry, dot relates to cos(angle)$$\mathbf{a}\cdot\mathbf{b}=\
Nonzero requirementCosine similarity divides by normsNeed \

Two common terminology notes #

  1. 1)Cosine similarity is the value cos⁡θ\cos\thetacosθ (range [−1, 1] in general).
  2. 2)People often define cosine distance as $1 - \mathrm{cosSim}(\mathbf{a},\mathbf{b})$. This is useful, but it is not guaranteed to be a metric on all vectors because the triangle inequality can fail. (It may behave metrically only under additional constraints, e.g., certain normalized nonnegative settings.)

If those boxes feel unfamiliar, you can still proceed—just revisit them when the formulas appear.

What Is Cosine Similarity? #

Why we need a “direction-only” similarity #

Suppose you have two vectors representing items:

Often, the direction encodes the “type” or “meaning,” while the length might reflect confidence, frequency, or just the model’s internal scaling.

If we used only the dot product a⋅b\mathbf{a}\cdot\mathbf{b}a⋅b to score similarity, then simply making vectors longer (larger magnitude) would inflate the score—even if the direction stayed the same. That can be undesirable when you want comparisons to be about alignment.

Cosine similarity fixes this by normalizing out the magnitudes.

Definition #

For two nonzero vectors a and b in ℝⁿ, cosine similarity is

cosSim(a,b)=a⋅b∥a∥ ∥b∥.\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}.cosSim(a,b)=∥a∥∥b∥a⋅b​.

Geometric meaning: “cosine of the angle” #

There is a key identity connecting dot product and angle:

a⋅b=∥a∥ ∥b∥cos⁡θ,\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}|,|\mathbf{b}|\cos\theta,a⋅b=∥a∥∥b∥cosθ,

where θ\thetaθ is the angle between a and b (in the usual Euclidean geometry). Rearranging gives

cos⁡θ=a⋅b∥a∥ ∥b∥=cosSim(a,b).\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|} = \mathrm{cosSim}(\mathbf{a},\mathbf{b}).cosθ=∥a∥∥b∥a⋅b​=cosSim(a,b).

So cosine similarity literally is the cosine of that angle.

Interpreting the value #

Because cosine ranges between −1 and 1:

In many embedding systems (especially after certain training setups), values are often mostly positive, but mathematically the full range [−1, 1] is possible.

Important constraint: nonzero vectors #

The formula divides by ∥a∥ ∥b∥|\mathbf{a}|,|\mathbf{b}|∥a∥∥b∥. If either vector is 0, then cosine similarity is undefined.

In practice, systems either:

Core Mechanic 1: Dot Product as Alignment (and why normalization matters) #

The dot product mixes direction and magnitude #

The dot product is

a⋅b=∑i=1naibi.\mathbf{a}\cdot\mathbf{b} = \sum_{i=1}^n a_i b_i.a⋅b=i=1∑n​ai​bi​.

It increases when:

  1. components match in sign and are large in magnitude, and

  2. the vectors point in similar directions.

But here’s the catch: if you scale a by a constant ccc, then

(ca)⋅b=c(a⋅b).(c\mathbf{a})\cdot\mathbf{b} = c(\mathbf{a}\cdot\mathbf{b}).(ca)⋅b=c(a⋅b).

So dot product is not scale-invariant.

A concrete example: same direction, inflated score #

Let

These all point in the same direction (45° line). Dot products:

The second looks “more similar” by dot product, but b and c are equally aligned with a—they just have different lengths.

Normalization removes the scale #

Cosine similarity divides by both lengths:

cosSim(a,b)=a⋅b∥a∥ ∥b∥.\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}.cosSim(a,b)=∥a∥∥b∥a⋅b​.

Now see what happens if we scale a by c>0c>0c>0:

\nShow your work (scale invariance)

Let a' = ca.

  1. Dot product scales:

a′⋅b=(ca)⋅b=c(a⋅b).\mathbf{a'}\cdot\mathbf{b} = (c\mathbf{a})\cdot\mathbf{b} = c(\mathbf{a}\cdot\mathbf{b}).a′⋅b=(ca)⋅b=c(a⋅b).

  1. Norm scales:

∥a′∥=∥ca∥=∣c∣ ∥a∥=c∥a∥(c>0).|\mathbf{a'}| = |c\mathbf{a}| = |c|,|\mathbf{a}| = c|\mathbf{a}| \quad (c>0).∥a′∥=∥ca∥=∣c∣∥a∥=c∥a∥(c>0).

  1. Plug into cosine similarity:

cosSim(a′,b)=c(a⋅b)(c∥a∥) ∥b∥=a⋅b∥a∥ ∥b∥.\mathrm{cosSim}(\mathbf{a'},\mathbf{b}) = \frac{c(\mathbf{a}\cdot\mathbf{b})}{(c|\mathbf{a}|),|\mathbf{b}|} = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}.cosSim(a′,b)=(c∥a∥)∥b∥c(a⋅b)​=∥a∥∥b∥a⋅b​.

So cosine similarity does not change when you scale one vector by a positive constant.

Unit vectors make the idea even clearer #

Define normalized (unit-length) vectors:

a^=a∥a∥,b^=b∥b∥.\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}, \quad \hat{\mathbf{b}} = \frac{\mathbf{b}}{|\mathbf{b}|}.a^=∥a∥a​,b^=∥b∥b​.

Then

cosSim(a,b)=a^⋅b^.\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = \hat{\mathbf{a}}\cdot\hat{\mathbf{b}}.cosSim(a,b)=a^⋅b^.

This is a powerful mental model:

Range and the Cauchy–Schwarz inequality #

A key guarantee is that cosine similarity is always between −1 and 1.

Cauchy–Schwarz says:

∣a⋅b∣≤∥a∥ ∥b∥.|\mathbf{a}\cdot\mathbf{b}| \le |\mathbf{a}|,|\mathbf{b}|.∣a⋅b∣≤∥a∥∥b∥.

Divide both sides by ∥a∥ ∥b∥|\mathbf{a}|,|\mathbf{b}|∥a∥∥b∥ (nonzero assumption):

∣a⋅b∥a∥ ∥b∥∣≤1.\left|\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}\right| \le 1.​∥a∥∥b∥a⋅b​​≤1.

So

−1≤cosSim(a,b)≤1.-1 \le \mathrm{cosSim}(\mathbf{a},\mathbf{b}) \le 1.−1≤cosSim(a,b)≤1.

This boundedness is one reason cosine similarity is numerically and conceptually convenient.

Core Mechanic 2: Angle, Sign, and What “Similarity” Really Means #

Cosine similarity is fundamentally about angle #

Because cosSim(a,b)=cos⁡θ\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = \cos\thetacosSim(a,b)=cosθ, it inherits the cosine curve’s behavior:

This is slightly different from many intuitive “distance” notions.

Negative cosine similarity: when opposite directions matter #

In some applications, negative similarity has a clear meaning:

In other applications (e.g., some retrieval systems), negative values might just be treated as “not similar” and thresholded away.

Relation to Euclidean distance (when vectors are normalized) #

If both vectors are normalized to unit length, then cosine similarity and Euclidean distance are tightly connected.

Let a^\hat{\mathbf{a}}a^ and b^\hat{\mathbf{b}}b^ be unit vectors. Consider squared Euclidean distance:

\nShow your work

∥a^−b^∥2=(a^−b^)⋅(a^−b^).|\hat{\mathbf{a}} - \hat{\mathbf{b}}|^2 = (\hat{\mathbf{a}} - \hat{\mathbf{b}})\cdot(\hat{\mathbf{a}} - \hat{\mathbf{b}}).∥a^−b^∥2=(a^−b^)⋅(a^−b^).

Expand:

=a^⋅a^−2a^⋅b^+b^⋅b^.= \hat{\mathbf{a}}\cdot\hat{\mathbf{a}} - 2\hat{\mathbf{a}}\cdot\hat{\mathbf{b}} + \hat{\mathbf{b}}\cdot\hat{\mathbf{b}}.=a^⋅a^−2a^⋅b^+b^⋅b^.

Since both are unit length:

a^⋅a^=1,b^⋅b^=1.\hat{\mathbf{a}}\cdot\hat{\mathbf{a}} = 1, \quad \hat{\mathbf{b}}\cdot\hat{\mathbf{b}} = 1.a^⋅a^=1,b^⋅b^=1.

So

∥a^−b^∥2=2−2(a^⋅b^)=2−2 cosSim(a,b).|\hat{\mathbf{a}} - \hat{\mathbf{b}}|^2 = 2 - 2(\hat{\mathbf{a}}\cdot\hat{\mathbf{b}}) = 2 - 2,\mathrm{cosSim}(\mathbf{a},\mathbf{b}).∥a^−b^∥2=2−2(a^⋅b^)=2−2cosSim(a,b).

Rearrange:

cosSim(a,b)=1−12∥a^−b^∥2.\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = 1 - \frac{1}{2}|\hat{\mathbf{a}} - \hat{\mathbf{b}}|^2.cosSim(a,b)=1−21​∥a^−b^∥2.

So on the unit sphere, cosine similarity is basically a monotonic transformation of Euclidean distance.

Cosine distance is common but not a guaranteed metric #

A frequently used “distance-like” quantity is

dcos⁡(a,b)=1−cosSim(a,b).d_{\cos}(\mathbf{a},\mathbf{b}) = 1 - \mathrm{cosSim}(\mathbf{a},\mathbf{b}).dcos​(a,b)=1−cosSim(a,b).

This has nice properties (0 when vectors match in direction, bigger when they differ), but it is not always a metric on ℝⁿ. In particular, the triangle inequality may fail.

Why that matters:

Practical takeaway: it’s fine to use $1-\cos$ as a heuristic distance for ranking and optimization, but don’t automatically assume metric guarantees unless you’ve checked the conditions for your specific setting.

Implementation note: numerical stability #

When vectors are very small or nearly zero, the denominator ∥a∥ ∥b∥|\mathbf{a}|,|\mathbf{b}|∥a∥∥b∥ can be tiny.

Common fix:

cosSimε(a,b)=a⋅bmax⁡(∥a∥ ∥b∥,ε)\mathrm{cosSim}_\varepsilon(\mathbf{a},\mathbf{b}) = \frac{\mathbf{a}\cdot\mathbf{b}}{\max(|\mathbf{a}|,|\mathbf{b}|,\varepsilon)}cosSimε​(a,b)=max(∥a∥∥b∥,ε)a⋅b​

or add ε inside the product. The exact choice depends on your numerical environment and expectations about zero vectors.

Application/Connection: Embeddings, Retrieval, and Attention Scoring #

1) Comparing embeddings (semantic similarity) #

Embeddings map discrete objects (words, items, images) to vectors. A core assumption is:

Cosine similarity is a natural fit because it ignores overall magnitude. This is especially helpful when vector norms vary due to:

Typical retrieval pipeline

  1. 1)Compute embedding for query q and for each candidate item xᵢ.
  2. 2)Score each candidate with cosSim(q,xi)\mathrm{cosSim}(\mathbf{q},\mathbf{x}_i)cosSim(q,xi​).
  3. 3)Return top-k.

Often, systems normalize all embeddings once (store unit vectors), making scoring just a dot product.

2) Cosine similarity as an attention score #

In attention mechanisms, we produce:

A general attention score between a query and key can be any similarity function. One simple option is cosine similarity:

s(q,k)=q⋅k∥q∥ ∥k∥.s(\mathbf{q},\mathbf{k}) = \frac{\mathbf{q}\cdot\mathbf{k}}{|\mathbf{q}|,|\mathbf{k}|}.s(q,k)=∥q∥∥k∥q⋅k​.

Then attention weights are typically:

αi=softmax(s(q,ki)).\alpha_i = \mathrm{softmax}(s(\mathbf{q},\mathbf{k}_i)).αi​=softmax(s(q,ki​)).

In modern transformers, the most common is scaled dot-product attention:

s(q,k)=q⋅kdk.s(\mathbf{q},\mathbf{k}) = \frac{\mathbf{q}\cdot\mathbf{k}}{\sqrt{d_k}}.s(q,k)=dk​​q⋅k​.

Why not always cosine?

That said, cosine attention variants exist and can be helpful in some regimes.

3) Practical tradeoffs: dot product vs cosine similarity #

ScoringFormulaSensitive to vector length?Common use
Dot producta·bYesMany attention layers; fast retrieval if norms are controlled
Cosine similarity(a·b)/(\a\\
Euclidean distance\a−b\

4) A workflow pattern: normalize once, then dot #

Because

cosSim(a,b)=a^⋅b^,\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = \hat{\mathbf{a}}\cdot\hat{\mathbf{b}},cosSim(a,b)=a^⋅b^,

a very common engineering trick is:

  1. store x^=x/∥x∥\hat{\mathbf{x}} = \mathbf{x}/|\mathbf{x}|x^=x/∥x∥ for every embedding

  2. compute similarity as a dot product

This can speed up retrieval (especially with vector databases / ANN indices) because dot products are highly optimized.

Worked Examples (3) #

Compute cosine similarity in 2D (with full arithmetic) #

Let a = (3, 4) and b = (4, 0). Compute cosSim(a, b) and interpret the result.

  1. Compute the dot product:

    \n$a⋅b=3⋅4+4⋅0=12.\mathbf{a}\cdot\mathbf{b} = 3\cdot 4 + 4\cdot 0 = 12.a⋅b=3⋅4+4⋅0=12.$

  2. Compute the norms:

    \n$∥a∥=32+42=9+16=5,|\mathbf{a}| = \sqrt{3^2 + 4^2} = \sqrt{9+16} = 5,∥a∥=32+42​=9+16​=5,$

    ∥b∥=42+02=16=4.|\mathbf{b}| = \sqrt{4^2 + 0^2} = \sqrt{16} = 4.∥b∥=42+02​=16​=4.

  3. Plug into cosine similarity:

    \n$cosSim(a,b)=125⋅4=1220=0.6.\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = \frac{12}{5\cdot 4} = \frac{12}{20} = 0.6.cosSim(a,b)=5⋅412​=2012​=0.6.$

  4. Interpretation:

    A cosine similarity of 0.6 means the vectors form an acute angle with moderate alignment (since 1 would be perfectly aligned, 0 would be perpendicular). If you want the angle explicitly:

    \n$θ=arccos⁡(0.6)≈53.13∘.\theta = \arccos(0.6) \approx 53.13^\circ.θ=arccos(0.6)≈53.13∘.$

Insight: Cosine similarity turned the raw dot product (12) into a bounded, scale-free score (0.6). The value directly corresponds to an angle, which is a clean geometric notion of similarity.

Magnitude invariance: two vectors with the same direction score equally #

Let q = (1, 2). Compare x₁ = (2, 4) and x₂ = (10, 20) by cosine similarity with q.

  1. Notice that x₁ = 2q and x₂ = 10q, so all three vectors point in the same direction.

  2. Compute cosSim(q, x₁):

    \nDot:

    q⋅x1=1⋅2+2⋅4=2+8=10.\mathbf{q}\cdot\mathbf{x}_1 = 1\cdot 2 + 2\cdot 4 = 2 + 8 = 10.q⋅x1​=1⋅2+2⋅4=2+8=10.

    Norms:

    ∥q∥=12+22=5,∥x1∥=22+42=20=25.|\mathbf{q}|=\sqrt{1^2+2^2}=\sqrt{5},\quad |\mathbf{x}_1|=\sqrt{2^2+4^2}=\sqrt{20}=2\sqrt{5}.∥q∥=12+22​=5​,∥x1​∥=22+42​=20​=25​.

    Cosine:

    cosSim(q,x1)=10(5)(25)=1010=1.\mathrm{cosSim}(\mathbf{q},\mathbf{x}_1)=\frac{10}{(\sqrt{5})(2\sqrt{5})}=\frac{10}{10}=1.cosSim(q,x1​)=(5​)(25​)10​=1010​=1.

  3. Compute cosSim(q, x₂) similarly:

    \nDot:

    q⋅x2=1⋅10+2⋅20=10+40=50.\mathbf{q}\cdot\mathbf{x}_2 = 1\cdot 10 + 2\cdot 20 = 10 + 40 = 50.q⋅x2​=1⋅10+2⋅20=10+40=50.

    Norms:

    ∥x2∥=102+202=500=105.|\mathbf{x}_2|=\sqrt{10^2+20^2}=\sqrt{500}=10\sqrt{5}.∥x2​∥=102+202​=500​=105​.

    Cosine:

    cosSim(q,x2)=50(5)(105)=5050=1.\mathrm{cosSim}(\mathbf{q},\mathbf{x}_2)=\frac{50}{(\sqrt{5})(10\sqrt{5})}=\frac{50}{50}=1.cosSim(q,x2​)=(5​)(105​)50​=5050​=1.

  4. Compare with dot products:

    \n$q⋅x1=10,q⋅x2=50.\mathbf{q}\cdot\mathbf{x}_1 = 10,\quad \mathbf{q}\cdot\mathbf{x}_2 = 50.q⋅x1​=10,q⋅x2​=50.$

    Dot product prefers x₂ purely because it is longer, while cosine similarity treats them as equally aligned.

Insight: Cosine similarity answers: “Do these vectors point the same way?” Dot product answers: “Are these vectors aligned and large?” That difference is exactly why cosine is popular for embedding comparisons.

Zero vector edge case (why the nonzero requirement exists) #

Let a = (0, 0, 0) and b = (1, −2, 3). Try to compute cosSim(a, b).

  1. Compute the dot product:

    \n$a⋅b=0⋅1+0⋅(−2)+0⋅3=0.\mathbf{a}\cdot\mathbf{b} = 0\cdot 1 + 0\cdot (-2) + 0\cdot 3 = 0.a⋅b=0⋅1+0⋅(−2)+0⋅3=0.$

  2. Compute the norms:

    \n$∥a∥=02+02+02=0,∥b∥=12+(−2)2+32=14.|\mathbf{a}| = \sqrt{0^2+0^2+0^2} = 0,\quad |\mathbf{b}| = \sqrt{1^2+(-2)^2+3^2} = \sqrt{14}.∥a∥=02+02+02​=0,∥b∥=12+(−2)2+32​=14​.$

  3. Plug into the formula:

    \n$cosSim(a,b)=00⋅14=00,\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = \frac{0}{0\cdot \sqrt{14}} = \frac{0}{0},cosSim(a,b)=0⋅14​0​=00​,$

    which is undefined.

  4. Practical resolution:

    If your system might produce zero vectors, you must decide on a policy: reject them, renormalize differently, or use an ε-stabilized denominator.

Insight: Cosine similarity is about direction, but the zero vector has no direction. The undefined division is not a nuisance—it reflects a real geometric ambiguity.

Key Takeaways #

Common Mistakes #

Practice #

easy

Compute cosSim(a, b) for a = (1, −1, 2) and b = (2, 0, 1).

Hint: Compute the dot product and each norm separately, then divide. Keep radicals until the end if you want exact form.

Show solution

Dot:

a⋅b=1⋅2+(−1)⋅0+2⋅1=2+0+2=4.\mathbf{a}\cdot\mathbf{b} = 1\cdot 2 + (-1)\cdot 0 + 2\cdot 1 = 2+0+2=4.a⋅b=1⋅2+(−1)⋅0+2⋅1=2+0+2=4.

Norms:

∥a∥=12+(−1)2+22=6,∥b∥=22+02+12=5.|\mathbf{a}|=\sqrt{1^2+(-1)^2+2^2}=\sqrt{6},\quad |\mathbf{b}|=\sqrt{2^2+0^2+1^2}=\sqrt{5}.∥a∥=12+(−1)2+22​=6​,∥b∥=22+02+12​=5​.

Cosine similarity:

cosSim(a,b)=465=430≈0.7303.\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = \frac{4}{\sqrt{6}\sqrt{5}} = \frac{4}{\sqrt{30}} \approx 0.7303.cosSim(a,b)=6​5​4​=30​4​≈0.7303.

medium

Show that cosSim(a, b) = cosSim(3a, 0.5b) for any nonzero vectors a, b.

Hint: Use how dot products and norms scale under scalar multiplication: (ca)·(db) and |ca|.

Show solution

Let a' = 3a and b' = 0.5b.

Dot scales:

a′⋅b′=(3a)⋅(0.5b)=1.5(a⋅b).\mathbf{a'}\cdot\mathbf{b'} = (3\mathbf{a})\cdot(0.5\mathbf{b}) = 1.5(\mathbf{a}\cdot\mathbf{b}).a′⋅b′=(3a)⋅(0.5b)=1.5(a⋅b).

Norms scale:

∥a′∥=∥3a∥=3∥a∥,∥b′∥=∥0.5b∥=0.5∥b∥.|\mathbf{a'}| = |3\mathbf{a}| = 3|\mathbf{a}|,\quad |\mathbf{b'}| = |0.5\mathbf{b}| = 0.5|\mathbf{b}|.∥a′∥=∥3a∥=3∥a∥,∥b′∥=∥0.5b∥=0.5∥b∥.

Cosine similarity:

cosSim(a′,b′)=1.5(a⋅b)(3∥a∥)(0.5∥b∥)=1.5(a⋅b)1.5∥a∥ ∥b∥=cosSim(a,b).\mathrm{cosSim}(\mathbf{a'},\mathbf{b'}) = \frac{1.5(\mathbf{a}\cdot\mathbf{b})}{(3|\mathbf{a}|)(0.5|\mathbf{b}|)} = \frac{1.5(\mathbf{a}\cdot\mathbf{b})}{1.5|\mathbf{a}|,|\mathbf{b}|} = \mathrm{cosSim}(\mathbf{a},\mathbf{b}).cosSim(a′,b′)=(3∥a∥)(0.5∥b∥)1.5(a⋅b)​=1.5∥a∥∥b∥1.5(a⋅b)​=cosSim(a,b).

hard

Assume |a| = |b| = 1 (unit vectors). If |a − b| = 0.8, compute cosSim(a, b).

Hint: Use the identity |a − b|² = 2 − 2(a·b) when both vectors are unit length.

Show solution

Given unit vectors, we have:

∥a−b∥2=2−2(a⋅b).|\mathbf{a}-\mathbf{b}|^2 = 2 - 2(\mathbf{a}\cdot\mathbf{b}).∥a−b∥2=2−2(a⋅b).

Compute squared distance:

∥a−b∥2=0.82=0.64.|\mathbf{a}-\mathbf{b}|^2 = 0.8^2 = 0.64.∥a−b∥2=0.82=0.64.

So:

0.64=2−2(a⋅b).0.64 = 2 - 2(\mathbf{a}\cdot\mathbf{b}).0.64=2−2(a⋅b).

Solve:

\n$$2(\mathbf{a}\cdot\mathbf{b}) = 2 - 0.64 = 1.36,$$

a⋅b=0.68.\mathbf{a}\cdot\mathbf{b} = 0.68.a⋅b=0.68.

Since unit vectors satisfy cosSim(a,b)=a⋅b\mathrm{cosSim}(\mathbf{a},\mathbf{b})=\mathbf{a}\cdot\mathbf{b}cosSim(a,b)=a⋅b,

cosSim(a,b)=0.68.\mathrm{cosSim}(\mathbf{a},\mathbf{b}) = 0.68.cosSim(a,b)=0.68.

Connections #

Related conceptual neighbors you may want next:

Quality: A (4.6/5)

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