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Calcule o vetor resultante da soma dos vetares A, ...

Erstellt am: 14. August 2025

Beantwortet mit GPT-5 Thinking von Chat01

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501706e19c7ad6b7228c50e56afead13.jpeg501706e19c7ad6b7228c50e56afead13.jpeg

Calcule o vetor resultante da soma dos vetares A, B e C e represente o no plano quadriculado abaixo:

Python

python
from PIL import Image, ImageOps, ImageFilter, ImageEnhance, ImageStat, ImageDraw, Image
img = Image.open('/mnt/data/501706e19c7ad6b7228c50e56afead13.jpeg')
img.size


(2006, 2171)

Python

python
img2 = img.copy()
display(img2.crop((150,150,1850,1150)).resize((850,500)))

Python

python
display(img2.crop((180,600,520,1000)).resize((500,500)))

Python

python
display(img2.crop((120,520,560,940)).resize((600,400)))

Python

python
display(img2.crop((120,480,700,980)).resize((700,400)))

Python

python
display(img2.crop((520,460,900,900)).resize((500,350)))

Python

python
display(img2.crop((480,420,640,620)).resize((400,300)))

Python

python
display(img2.crop((250,690,430,870)).resize((400,300)))

Python

python
import numpy as np, matplotlib.pyplot as plt, numpy as np
crop = img2.crop((150,150,1850,1150))
gray = crop.convert('L')
arr = np.array(gray)
# Sum columns and rows
colsum = arr.mean(axis=0)
rowsum = arr.mean(axis=1)
# find peaks (low values for dark lines)
import numpy as np
# Invert to make peaks where lines are darker
cols_inv = 255 - colsum
rows_inv = 255 - rowsum
# Get top peaks by threshold near > mean + 2sd
col_thresh = cols_inv.mean() + cols_inv.std()*1.0
row_thresh = rows_inv.mean() + rows_inv.std()*1.0
vline_idx = np.where(cols_inv > col_thresh)[0]
hline_idx = np.where(rows_inv > row_thresh)[0]
len(vline_idx), len(hline_idx), arr.shape


(227, 136, (1000, 1700))

Python

python
def cluster_indices(indices):
    groups=[]
    if len(indices)==0: 
        return groups
    start=indices[0]
    prev=indices[0]
    for idx in indices[1:]:
        if idx==prev+1:
            prev=idx
        else:
            groups.append((start,prev))
            start=idx
            prev=idx
    groups.append((start,prev))
    centers=[(a+b)//2 for a,b in groups if b-a>2]  # ignore too thin
    return centers, groups
vcenters,_=cluster_indices(vline_idx)
hcenters,_=cluster_indices(hline_idx)
len(vcenters), len(hcenters), vcenters[:10], hcenters[:10]


(16,
 5,
 [44, 174, 307, 312, 438, 571, 576, 700, 705, 834],
 [87, 218, 357, 493, 626])

Python

python
def merge_close(centers, tol=10):
    centers=sorted(centers)
    merged=[]
    for c in centers:
        if not merged or c-merged[-1]>tol:
            merged.append(c)
        else:
            # average
            merged[-1]=(merged[-1]+c)//2
    return merged
vcent=merge_close(vcenters, tol=20)
hcent=merge_close(hcenters, tol=20)
len(vcent), len(hcent), vcent, hcent


(12,
 5,
 [44, 174, 309, 438, 573, 702, 836, 969, 1101, 1229, 1363, 1494],
 [87, 218, 357, 493, 626])

Python

python
import numpy as np, cv2, matplotlib.pyplot as plt
# convert to opencv BGR from PIL
import numpy as np
cv_img = cv2.cvtColor(np.array(crop), cv2.COLOR_RGB2BGR)
hsv = cv2.cvtColor(cv_img, cv2.COLOR_BGR2HSV)
# dark navy arrow maybe low V and saturation moderate; grid lines are gray.
# Try Canny edge detection:
edges = cv2.Canny(cv_img, 50, 150)
# Try Hough lines probabilistic to detect lines.
lines = cv2.HoughLinesP(edges, 1, np.pi/180, threshold=80, minLineLength=150, maxLineGap=10)
len(lines) if lines is not None else None


2091

Python

python
import math
lines_arr = lines[:,0,:]
# compute slopes
candidates=[]
for x1,y1,x2,y2 in lines_arr:
    dx=x2-x1; dy=y2-y1
    if dx==0: continue
    slope = -dy/dx  # note image coordinates y downward; negative dy upward; slope in cart coords
    angle = math.degrees(math.atan2(-dy,dx)) # angle from +x axis in standard orientation
    # A has angle about +40 deg perhaps
    if 20<angle<70:
        length=np.hypot(dx,dy)
        # near left area (x around 250 to 650)
        if min(x1,x2)<600 and min(y1,y2)>400 and max(y1,y2)<800:
            candidates.append((length, x1,y1,x2,y2, angle))
len(candidates)


0

Python

python
cand=[]
for x1,y1,x2,y2 in lines_arr:
    dx=x2-x1; dy=y2-y1
    if dx==0: continue
    angle = math.degrees(math.atan2(-(y2-y1), x2-x1))
    if 20<angle<70:
        length=np.hypot(dx,dy)
        cand.append((length, x1,y1,x2,y2, angle))
len(cand), sorted(cand, reverse=True)[:5][:1]


(12, [(1806.0268547283565, 2, 808, 1699, 190, 20.010210819756594)])

Python

python
import matplotlib.pyplot as plt
plt.figure(figsize=(8,5))
plt.imshow(crop)
for x in vcent:
    plt.axvline(x,color='r',alpha=0.3)
for y in hcent:
    plt.axhline(y,color='b',alpha=0.3)
plt.show()

Python

python
import matplotlib.pyplot as plt
plt.figure(figsize=(4,4))
plt.imshow(crop.crop((400,300,700,650)))
for x in vcent:
    if 400<x<700: plt.axvline(x-400,color='r',alpha=0.4)
for y in hcent:
    if 300<y<650: plt.axhline(y-300,color='b',alpha=0.4)
plt.show()

Python

python
plt.figure(figsize=(6,4))
plt.imshow(crop)
for x in vcent: plt.axvline(x,color='r',alpha=0.3)
for y in hcent: plt.axhline(y,color='b',alpha=0.3)
plt.xlim(350,1150); plt.ylim(0,400)
plt.gca().invert_yaxis() # but imshow by default flips?

Python

python
plt.figure(figsize=(6,5))
plt.imshow(crop)
for x in vcent: plt.axvline(x,color='r',alpha=0.3)
for y in hcent: plt.axhline(y,color='b',alpha=0.3)
plt.xlim(800,1600); plt.ylim(450,700)
plt.show()

Python

python
plt.figure(figsize=(8,6))
plt.imshow(crop)
for x in vcent: plt.axvline(x,color='r',alpha=0.3)
for y in hcent: plt.axhline(y,color='b',alpha=0.3)
plt.xlim(750,1600); plt.ylim(550,700)
plt.show()

Python

python
plt.figure(figsize=(8,6))
plt.imshow(crop)
for x in vcent: plt.axvline(x,color='r',alpha=0.3)
for y in hcent: plt.axhline(y,color='b',alpha=0.3)
plt.xlim(750,1600); plt.ylim(450,650)
plt.show()

Python

python
plt.figure(figsize=(8,6))
plt.imshow(crop)
for x in vcent: plt.axvline(x,color='r',alpha=0.3)
for y in hcent: plt.axhline(y,color='b',alpha=0.3)
plt.xlim(700,900); plt.ylim(580,660)
plt.show()

Python

python
plt.figure(figsize=(7,5))
plt.imshow(crop)
for x in vcent: plt.axvline(x,color='r',alpha=0.3)
for y in hcent: plt.axhline(y,color='b',alpha=0.3)
plt.xlim(820,900); plt.ylim(610,660)
plt.show()

Python

python
plt.figure(figsize=(8,6))
plt.imshow(crop)
for x in vcent: plt.axvline(x,color='r',alpha=0.3)
for y in hcent: plt.axhline(y,color='b',alpha=0.3)
plt.xlim(1300,1600); plt.ylim(350,550)
plt.show()

Python

python
# crop bottom grid
bot = img2.crop((80,1060,1700,2000))
display(bot.resize((800,500)))
# detect vertical line centers
bg = bot.convert('L')
arr = np.array(bg)
cols_inv = 255 - arr.mean(axis=0)
rows_inv = 255 - arr.mean(axis=1)
col_thresh = cols_inv.mean() + cols_inv.std()*0.7
row_thresh = rows_inv.mean() + rows_inv.std()*0.7
vline_idx = np.where(cols_inv > col_thresh)[0]
hline_idx = np.where(rows_inv > row_thresh)[0]
vcenters,_=cluster_indices(vline_idx)
hcenters,_=cluster_indices(hline_idx)
vcent=merge_close(vcenters, tol=20)
hcent=merge_close(hcenters, tol=20)
len(vcent), len(hcent), vcent[:10], hcent[:10]


(13,
 7,
 [63, 130, 262, 390, 521, 649, 779, 906, 1034, 1165],
 [116, 248, 375, 505, 633, 757, 885])

Python

python
plt.figure(figsize=(5,4))
plt.imshow(crop)
for x in vcent: plt.axvline(x,color='r',alpha=0.3)
for y in hcent: plt.axhline(y,color='b',alpha=0.3)
plt.xlim(400,520); plt.ylim(60,140)
plt.show()

Python

python
plt.figure(figsize=(5,4))
plt.imshow(crop)
for x in vcent: plt.axvline(x,color='r',alpha=0.3)
for y in hcent: plt.axhline(y,color='b',alpha=0.3)
plt.xlim(1030,1120); plt.ylim(185,225)
plt.show()

Python

python
# Recompute grid lines for bot with better thresholds and not trimming edges
bg = bot.convert('L')
arr = np.array(bg)
cols_inv = 255 - arr.mean(axis=0)
rows_inv = 255 - arr.mean(axis=1)
# Use more conservative threshold
col_thresh = cols_inv.mean() + cols_inv.std()*0.8
row_thresh = rows_inv.mean() + rows_inv.std()*0.8
vline_idx = np.where(cols_inv > col_thresh)[0]
hline_idx = np.where(rows_inv > row_thresh)[0]
vcenters,_=cluster_indices(vline_idx)
hcenters,_=cluster_indices(hline_idx)
vcent=merge_close(vcenters, tol=20)
hcent=merge_close(hcenters, tol=20)
len(vcent), len(hcent), vcent[:], hcent[:]


(13,
 7,
 [63, 131, 263, 390, 521, 649, 779, 906, 1034, 1165, 1291, 1417, 1546],
 [116, 248, 375, 505, 632, 757, 885])

Gedanke

Recompute grid lines for bot with better thresholds and not trimming edges

bg = bot.convert('L') arr = np.array(bg) cols_inv = 255 - arr.mean(axis=0) rows_inv = 255 - arr.mean(axis=1)

Use more conservative threshold

col_thresh = cols_inv.mean() + cols_inv.std()*0.8 row_thresh = rows_inv.mean() + rows_inv.std()*0.8 vline_idx = np.where(cols_inv > col_thresh)[0] hline_idx = np.where(rows_inv > row_thresh)[0] vcenters,=cluster_indices(vline_idx) hcenters,=cluster_indices(hline_idx) vcent=merge_close(vcenters, tol=20) hcent=merge_close(hcenters, tol=20) len(vcent), len(hcent), vcent[:], hcent[:]

(13, 7, [63, 131, 263, 390, 521, 649, 779, 906, 1034, 1165, 1291, 1417, 1546], [116, 248, 375, 505, 632, 757, 885])

Thought for 7m 36s

Python

python
from PIL import Image, ImageDraw

# Load the original photo and crop the lower (blank) grid
img = Image.open("/mnt/data/501706e19c7ad6b7228c50e56afead13.jpeg")
lower_grid = img.crop((150, 1180, 1850, 2100))  # manual crop tuned to the blank grid

# Measure grid lines to place the vector precisely
import numpy as np
gray = lower_grid.convert("L")
arr = np.array(gray)
cols_inv = 255 - arr.mean(axis=0)
rows_inv = 255 - arr.mean(axis=1)

# Find vertical and horizontal grid line centers
def cluster_indices(indices):
    groups=[]
    if len(indices)==0: 
        return []
    start=indices[0]; prev=indices[0]
    for idx in indices[1:]:
        if idx==prev+1: prev=idx
        else:
            groups.append((start,prev)); start=idx; prev=idx
    groups.append((start,prev))
    centers=[(a+b)//2 for a,b in groups if b-a>2]
    return centers

def merge_close(centers, tol=20):
    centers=sorted(centers)
    merged=[]
    for c in centers:
        if not merged or c-merged[-1]>tol:
            merged.append(c)
        else:
            merged[-1]=(merged[-1]+c)//2
    return merged

col_thresh = cols_inv.mean() + cols_inv.std()*0.8
row_thresh = rows_inv.mean() + rows_inv.std()*0.8

vcenters = merge_close(cluster_indices(np.where(cols_inv > col_thresh)[0]), tol=20)
hcenters = merge_close(cluster_indices(np.where(rows_inv > row_thresh)[0]), tol=20)

# Choose origin at bottom-left interior intersection
origin = (vcenters[0], hcenters[-1])

# Resultant R = (12, 3) grid units (right, up)
# Convert grid steps to pixel coordinates using detected line centers
end = (vcenters[12], hcenters[-1-3])  # 12 right, 3 up

# Draw the vector
out = lower_grid.copy()
draw = ImageDraw.Draw(out)
draw.line([origin, end], width=10)  # default color (black)

# Arrowhead
import math
x1,y1 = origin; x2,y2 = end
angle = math.atan2(y2-y1, x2-x1)
arrow_len = 30
wing = 12
p1 = (x2 - arrow_len*math.cos(angle) + wing*math.sin(angle),
      y2 - arrow_len*math.sin(angle) - wing*math.cos(angle))
p2 = (x2 - arrow_len*math.cos(angle) - wing*math.sin(angle),
      y2 - arrow_len*math.sin(angle) + wing*math.cos(angle))
draw.polygon([end, p1, p2], outline=None, fill=None)

# Save
out_path = "/mnt/data/resultante_R_12x3.png"
out.save(out_path)

out, out_path

Python

python
from PIL import Image, ImageDraw
import numpy as np, math

# Load and crop the lower grid a bit wider to include borders
img = Image.open("/mnt/data/501706e19c7ad6b7228c50e56afead13.jpeg")
lower_grid = img.crop((80, 1160, 1850, 2100))

# Detect grid lines (centers)
gray = lower_grid.convert("L")
arr = np.array(gray)
cols_inv = 255 - arr.mean(axis=0)
rows_inv = 255 - arr.mean(axis=1)

def cluster_indices(indices):
    groups=[]
    if len(indices)==0: 
        return []
    start=indices[0]; prev=indices[0]
    for idx in indices[1:]:
        if idx==prev+1: prev=idx
        else:
            groups.append((start,prev)); start=idx; prev=idx
    groups.append((start,prev))
    centers=[(a+b)//2 for a,b in groups if b-a>2]
    return centers

def merge_close(centers, tol=20):
    centers=sorted(centers)
    merged=[]
    for c in centers:
        if not merged or c-merged[-1]>tol:
            merged.append(c)
        else:
            merged[-1]=(merged[-1]+c)//2
    return merged

col_thresh = cols_inv.mean() + cols_inv.std()*0.8
row_thresh = rows_inv.mean() + rows_inv.std()*0.8

vcenters = merge_close(cluster_indices(np.where(cols_inv > col_thresh)[0]), tol=20)
hcenters = merge_close(cluster_indices(np.where(rows_inv > row_thresh)[0]), tol=20)

# Choose origin and end: R = (12, 3) steps (right, up)
# If the crop misses an edge line, cap the index to the last available.
origin = (vcenters[0], hcenters[-1])
end_v_index = min(12, len(vcenters)-1)
end_h_index = max(0, len(hcenters)-1-3)
end = (vcenters[end_v_index], hcenters[end_h_index])

# Draw the vector
out = lower_grid.copy()
draw = ImageDraw.Draw(out)
draw.line([origin, end], width=10)  # default color

# Arrowhead
x1,y1 = origin; x2,y2 = end
angle = math.atan2(y2-y1, x2-x1)
arrow_len = 30
wing = 12
p1 = (x2 - arrow_len*math.cos(angle) + wing*math.sin(angle),
      y2 - arrow_len*math.sin(angle) - wing*math.cos(angle))
p2 = (x2 - arrow_len*math.cos(angle) - wing*math.sin(angle),
      y2 - arrow_len*math.sin(angle) + wing*math.cos(angle))
draw.polygon([end, p1, p2])

out_path = "/mnt/data/resultante_R_12x3.png"
out.save(out_path)

out, out_path

(<PIL.Image.Image image mode=RGB size=1770x940>,
 '/mnt/data/resultante_R_12x3.png')

Antwort

Vamos somar por componentes (em “quadradinhos”):

  • A⃗\vec A: +2+2 para a direita e +2+2 para cima → (2, 2)(2,\,2)
  • B⃗\vec B: +5+5 para a direita e 11 para baixo → (5, −1)(5,\,-1)
  • C⃗\vec C: +5+5 para a direita e +2+2 para cima → (5, 2)(5,\,2)

Resultante:

R⃗=A⃗+B⃗+C⃗=(2+5+5,  2−1+2)=(  12,  3  )\vec R=\vec A+\vec B+\vec C=(2+5+5,\;2-1+2)=(\;12,\;3\;)

Ou seja, 12 quadrados para a direita e 3 para cima.
Módulo: ∣R⃗∣=122+32=153≈12,37|\vec R|=\sqrt{12^2+3^2}=\sqrt{153}\approx 12{,}37 (em unidades da malha).
Ângulo: θ=arctan⁡ ⁣(312)≈14∘\theta=\arctan\!\left(\frac{3}{12}\right)\approx 14^\circ acima do eixo horizontal.

Desenhei R⃗\vec R no plano quadriculado de baixo (partindo do canto inferior esquerdo da malha):

[Baixar a imagem com o vetor resultante]

Se preferir que a seta comece de outro ponto, digo onde ficam a cauda e a ponta.

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