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Vektor · Tingkatan 4

Arah dan Magnitud Vektor Unit dalam Segi Empat Selari

Direction and Magnitude of a Unit Vector in a Parallelogram

Direction and Magnitude of a Unit Vector in a Parallelogram

Soalan

The question

(b) Diberi KLMN ialah sebuah segi empat selari, dengan keadaan LM⃗=ḭ-3j̰ dan MN⃗=-3ḭ+4j̰. Cari vektor unit dalam arah KM⃗.
Rajah 7 menunjukkan satah Cartes dengan paksi-x dan paksi-y bersilang pada asalan O. Vektor OP⃗ dilukis ke titik P(4, 4) di sukuan pertama, dan vektor OQ⃗ dilukis ke titik Q(6, -4) di sukuan keempat.

Untuk mencari vektor unit dalam arah KM⃗, cari dahulu vektor paduan KM⃗ = KL⃗ + LM⃗ dengan menambah komponen secara betul untuk mendapatkan 4ḭ - 7j̰. Bahagikan vektor ini dengan magnitudnya, |KM⃗| = √(4² + (-7)²) = √65. Jawapan akhir ialah 1/(√65)(4ḭ - 7j̰).

To find the unit vector in the direction of KM⃗, first find the resultant vector KM⃗ = KL⃗ + LM⃗ by adding components correctly to get 4ḭ - 7j̰. Divide this vector by its magnitude, |KM⃗| = √(4² + (-7)²) = √65. The final answer is 1/(√65)(4ḭ - 7j̰).

Kertas sebenar, sudah ditanda

The marked scripts

Working stops at the resultant vector KM without calculating the unit vector in the direction of KM.
Pengiraan berhenti di vektor paduan KM tanpa mencari vektor unit dalam arah KM. Working stops at the resultant vector KM without calculating the unit vector in the direction of KM.
An arithmetic error where (-7) squared is calculated as 44 instead of 49 inside the square root.
Kesilapan aritmetik pada pengiraan kuasa dua (-7) yang ditulis sebagai 44 dan bukannya 49. An arithmetic error where (-7) squared is calculated as 44 instead of 49 inside the square root.
The negative sign of the j component was accidentally changed to a positive sign in the final answer.
Tanda negatif bagi pengkali j secara tidak sengaja ditukar menjadi positif pada jawapan akhir. The negative sign of the j component was accidentally changed to a positive sign in the final answer.
Vector LM was subtracted instead of added when determining the resultant vector KM.
Vektor LM ditolak bukannya ditambah semasa menentukan vektor paduan KM. Vector LM was subtracted instead of added when determining the resultant vector KM.

Yang salah

What went wrong

KM⃗ = KL⃗ + LM⃗
= (3ḭ - 4j̰) + (ḭ - 3j̰)
= 4ḭ - 7j̰
KM̂ = (4ḭ - 7j̰)/(√(4² + (-7)²))
= (4ḭ - 7j̰)/(√(16 + 44))

Yang betul

The correct working

KL⃗ = -MN⃗ = -(-3ḭ + 4j̰) = 3ḭ - 4j̰
KM⃗ = KL⃗ + LM⃗
= (3ḭ - 4j̰) + (ḭ - 3j̰)
= 4ḭ - 7j̰
|KM⃗| = √(4² + (-7)²) = √(16 + 49) = √65
KM̂ = 1/(√65)(4ḭ - 7j̰)

Kenapa ramai buat silap ini

Why this happens

Kesilapan kerap berlaku apabila menentukan tanda arah bagi vektor selari dalam segi empat selari, atau ketika menghitung kuasa dua nombor negatif seperti (-7)² = 49. Selain itu, ada pelajar yang terlupa membahagi dengan magnitud atau tersilap menukar tanda negatif komponen j̰ kepada positif pada langkah akhir.

Errors often occur when determining the direction signs for parallel vectors in a parallelogram, or when squaring a negative number such as (-7)² = 49. Additionally, some students forget to divide by the magnitude or accidentally change the negative sign of the j̰ component to positive in the final step.

Markah yang hilang

Marks lost

3 markah

3 marks

Chegu Khairi menanda kerja ini setiap minggu

Chegu Khairi marks this every week

Kerja rumah ditanda pada hari yang sama dan dibincangkan dalam kelas berikutnya, jadi kesalahan sebegini dapat diperbetulkan semasa latihan dan bukan berlaku semasa peperiksaan.

Homework is marked the same day and discussed in the next class, so a mistake like this one gets corrected during practice rather than happening in the exam.

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