Vector Operations

Number Theory and Algebra

• Vector Operations
  • Vector addition
  • Vector angles
  • Vector cross product
  • Vector dot product

Source: HP-45 Applications Book (HP 00045-90320 Rev. B Reorder 00045-66001, Dec 1974)
Source: HP 35s User's Guide HP part number F2215AA-90001

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Vector Operations

Vector addition

Note:

The HP 35s cannot handle vectors with more than 3 dimensions.

Suppose vector V_k (in 2-dimensional space) has magnitude m_k and direction θ_k (k = 1,2, …, n). Find the sum

Example:

mk	θk
2	30°
6.2	-45°
7.6	125°
10.7	232°

Answer:

V = -4.83i⃗ - 5.59j⃗

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1		DISPLAY 9'xiy		Display Mode x+𝒾y
2	mk			Perform 2-3 for k=1,2,…,n
3	θk	+	Vk	displayed as x+𝒾y

Vector angles

Suppose

x⃗ = (x₁, x₂, x₃)
y⃗ = (y₁, y₂, y₃)

then the angle between these two vecors is

θ = arccos [ ( x⃗ · y⃗ ) ÷ ( | x⃗ || y⃗ | ) ] = arccos [ (x₁y₁ + x₂y₂ + x₃y₃) ÷ ( √(x₁²+x₂²+x₃²) √(y₁²+y₂²+y₃²) ) ]

Example:

Find the angle between

x⃗ = (5, -6.2, -7),
y⃗ = (3.15, 2.22, -0.3)

Answer:

θ= 84.28 degrees = 1.47 radians = 93.64 grads.

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x	ENTER
2	y	ENTER ENTER 1 × R↓		Copy y and x into Z and T
3		×	xy	Dot product of x⃗ and y⃗
4		x<>y ABS	Mag	Absolute value of x⃗
5		÷ x<>y ABS	Mag	Absolute value of y⃗
6		÷ ACOS	Ang	Angle θ

; XEQ \Gh ENTER
LBL G
  ABS
  CLx
  eqn 'ACOS(LASTx*REGY/ABS(LASTx)/ABS(REGY))'
RTN

Vector cross product

Formula:

If x⃗ = (x₁, x₂, x₃) and y⃗ = (y₁, y₂, y₃) are two vectors, then cross product z⃗ is also a vector.

z⃗ = x⃗ ⨯ y⃗
= (x₂y₃ - x₃y₂, x₃y₁ - x₁y₃, x_ay₂ - x₂y₁)
= (z₁, z₂, z₃)

Example:

If

x⃗ = (2.34, 5,17, 7.43),
y⃗ = (0.072, 0.231, 0.409)

Find x⃗ ⨯ y⃗

Answer:

x⃗ ⨯ y⃗ = (0.40, -0.42, 0.17)

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x2	STO B
2	y3	STO Y ×
3	x3	STO X
4	y2	STO D × -	z1
5	y1	STO C RCL ×
6		X
7	x1	STO A RCL ×
8		Y -	z2
9		RCL A RCL × D
10		RCL B RCL × C
11		-	z3

; XEQ 1 ENTER
LBL X
  ABS
  CLx
  eqn 'LASTx*[1,0,0]\|>C'
  CLx
  eqn 'LASTx*[0,1,0]\|>D'
  CLx
  eqn 'LASTx*[0,0,1]\|>Y'
  CLx
  eqn 'REGY*[1,0,0]\|>A'
  CLx
  eqn 'REGY*[0,1,0]\|>B'
  CLx
  eqn 'REGY*[0,0,1]\|>X'
  CLx
  eqn '[B*Y-X*D,X*C-A*Y,A*D-B*C]'
RTN

Vector dot product

Note:

The HP 35s cannot handle vectors with more than 3 dimensions.

Formulas:

Given two vectors x⃗ , y⃗ an n-dimensional vector space

x⃗ = (x₁, x₂, …, x_n)
y⃗ = (y₁, y₂, …, y_n)

the dot product is

x⃗ · y⃗ = x₁y₁ + x₂y₂ + … + x_ny_n

Example:

If

x⃗ = (2.34, 5.17, 7.43, 9.11, 11.41),
y⃗ = (0.072, 0.231, 0.409, 0.703, 0.891)

then x⃗ · y⃗ = 20.97

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1		CLEAR 4'Σ
2	yi	ENTER		Perform 2-3 for i=1,2,…,n
3	xi	Σ+	i
4		SUMS 6'Σxy