Calculate the vector component

Question

Challenge

Assume two vectors \$\mathbf{a} = (a_1,a_2,\cdots,a_n)\$ and \$\mathbf{b} = (b_1,b_2,\cdots,b_n)\$ are given in an \$n\$-dimensional space, where at least one of \$b_1,\cdots,b_n\$ is nonzero. Then \$\mathbf{a}\$ can be uniquely decomposed into two vectors, one being a scalar multiple of \$\mathbf{b}\$ and one perpendicular to \$\mathbf{b}\$:

$$ \mathbf{a} = \mathbf{b}x + \mathbf{b^\perp}\text{, where }\mathbf{b^\perp} \cdot \mathbf{b}=0. $$

Given \$\mathbf{a}\$ and \$\mathbf{b}\$ as input, find the value of \$x\$.

This can be also thought of as the following: Imagine a line passing through the origin and the point \$\mathbf{b}\$. Then draw a perpendicular line on it that passes through the point \$\mathbf{a}\$, and denote the intersection \$\mathbf{c}\$. Finally, find the value of \$x\$ that satisfies \$\mathbf{c}=\mathbf{b}x\$.

You can use an explicit formula too (thanks to @xnor), which arises when calculating the projection:

$$ x=\frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} $$

Standard code-golf rules apply. The shortest code in bytes wins.

Example

Here is an example in 2D space, where a=(2,7) and b=(3,1). Observe that (2,7) = (3.9,1.3) + (-1.9,5.7) where (3.9,1.3) is equal to 1.3b and (-1.9,5.7) is perpendicular to b. Therefore, the expected answer is 1.3.

Test cases

a         b          answer
(2,7)     (3,1)      1.3
(2,7)     (-1,3)     1.9
(3,4,5)   (0,0,1)    5
(3,4,5)   (1,1,1)    4
(3,4,5)   (1,-1,-1)  -2
(3,4,5,6) (1,-2,1,2) 1.2

Answer 1

APL (Dyalog Unicode), 1 byte^SBCS

⌹

Check all test cases! When used dyadically, X ⌹ Y solves the least squares^* system \$Ya = X\$ for a result \$a\$ of the appropriate shape, e.g.:

• if \$Y\$ is a matrix and \$X\$ is a vector, we try to solve a linear system of equations.
• if \$Y\$ and \$X\$ are matrices, we compute \$Y\$'s (pseudo-)inverse and multiply it on the left of \$X\$.
• when both \$X\$ and \$Y\$ are vectors, the least squares formulation reduces to what we want, namely

$$\frac{X \cdot Y}{||Y||^2}$$

^*the least squares system \$Ya = X\$ can be understood as "what should \$a\$ be such that \$Ya\$ is as close as can be to \$X\$?", where closeness is measured with the usual L2 distance.

Answer 2

Haskell, 33 32 bytes

-1 byte thanks to @xnor!

(!)b=sum.zipWith(*)b
a#b=a!b/b!b

Try it online!

---

Coconut, 35 bytes

(a,b)->p(a,b)/p(b,b)
p=sum..map$(*)

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Answer 3

R, 24 bytes

function(a,b)a%*%b/b%*%b

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Answer 4

Python 3 + numpy, 20 bytes

lambda a,b:a@b/(b@b)

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Answer 5

MATL, 2 bytes

Y\

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Least squares approach such as used in the APL answer.

Answer 6

Jelly, 6 5 bytes

ḋ`÷@ḋ

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Simple translation of the given formula. Takes \$\mathbf{b}\$ as the left argument and \$\mathbf{a}\$ as the right argument.

ḋ        The dot product of b and
 `       itself,
  ÷@     dividing
    ḋ    the dot product of b and a.

Answer 7

Charcoal, 21 18 bytes

Ｆ²⊞υΣＥＡ×κ§θλＩ∕⊟υ⊟υ

Try it online! Link is to verbose version of code. Takes inputs in the order b, a. Explanation:

Ｆ²

Repeat twice...

⊞υΣＥＡ×κ§θλ

Input a vector, take its dot product with b and push the result to the predefined empty list.

Ｉ∕⊟υ⊟υ

Retrieve the dot products and take their quotient.

Answer 8

05AB1E, 6 bytes

*OInO/

Try it online or verify all test cases.

Implements the given formula:

$$x = \frac{a_1\times b_1 + a_2\times b_2 + \dots + a_n\times b_n}{b_1^2 + b_2^2 + \dots + b_n^2}$$

Explanation:

*       # Multiply the values at the same indices in the two (implicit) input-lists
 O      # Sum this list
  I     # Push the second input-list again
   n    # Square each value
    O   # Take the sum of that
     /  # And divide the two values
        # (after which the result is output implicitly)

Answer 9

C (gcc), ^{84 74} 73 bytes

Saved 10 bytes thanks to dingledooper!!!

Saved a byte thanks to ceilingcat!!!

float f(a,b,n)float*a,*b;{float x,y;for(;n--;y+=*b**b++)x+=*a++**b;x/=y;}

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Inputs two pointers to vectors \$a,b\$ and their dimension \$n\$ and returns their component..

Uses given formula:

$$x = \frac{a_0\cdot b_0 + a_1\cdot b_1 + \dots + a_{n-1}\cdot b_{n-1}}{b_0^2 + b_1^2 + \dots + b_{n-1}^2}$$

Answer 10

Wolfram Language (Mathematica), 9 bytes

#.#2/#.#&

Try it online! Pure function. Takes b followed by a as input and returns a rational number as output. It just directly uses Mathematica's notation for the dot product.

Answer 11

Fortran >= 95, 66 bytes

Taking advantage of implicit typing for the return type.

function x(a,b)
real a(:),b(:)
x=dot_product(a,b)/norm2(b)**2
end

Answer 12

[R], 22 bytes

Taking advantage of a resident regression function, lm

function(a,b)lm(a~b-1)

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Answer 13

Java 10, 84 74 bytes

a->b->{float A=0,B=0;int i=0;for(var t:b){A+=a[i++]*t;B+=t*t;}return A/B;}

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Answer 14

Japt v2.0a0, 9 bytes

í*V x÷Vx²

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