7. Koordinat bagi titik $$ A $$ dan titik $$ B $$ masing-masing ialah $$ (3,1) $$ dan $$ (2 h, h) $$. Diberi $$ A B $$ ialah vektor unit. KBAT Dengan menggunakan hukum segi tiga vektor, cari nilai-nilai yang mungkin bagi $$ h $$. SPON The coordinates of points $$ A $$ and $$ B $$ are $$ (3,1) $$ and $$ (2 h, h) $$ respectively. Given $$ \overrightarrow{A B} $$ is a unit vector. Using the vector's triangle law, find the possible values of $$ h $$.

Question

UpStudy AI · Accepted Answer

To solve the problem, we need to find the possible values of $$ h $$ such that the vector $$ \overrightarrow{AB} $$ is a unit vector.

### Step 1: Find the vector $$ \overrightarrow{AB} $$

The vector $$ \overrightarrow{AB} $$ can be calculated using the coordinates of points $$ A $$ and $$ B $$:

$$
\overrightarrow{AB} = B - A = (2h - 3, h - 1)
$$

### Step 2: Calculate the magnitude of $$ \overrightarrow{AB} $$

The magnitude of a vector $$ (x, y) $$ is given by the formula:

$$
\text{Magnitude} = \sqrt{x^2 + y^2}
$$

For our vector $$ \overrightarrow{AB} = (2h - 3, h - 1) $$, the magnitude is:

$$
\|\overrightarrow{AB}\| = \sqrt{(2h - 3)^2 + (h - 1)^2}
$$

### Step 3: Set the magnitude equal to 1

Since $$ \overrightarrow{AB} $$ is a unit vector, we set its magnitude equal to 1:

$$
\sqrt{(2h - 3)^2 + (h - 1)^2} = 1
$$

### Step 4: Square both sides to eliminate the square root

Squaring both sides gives:

$$
(2h - 3)^2 + (h - 1)^2 = 1
$$

### Step 5: Expand the equation

Now we expand both squares:

$$
(2h - 3)^2 = 4h^2 - 12h + 9
$$
$$
(h - 1)^2 = h^2 - 2h + 1
$$

Combining these, we have:

$$
4h^2 - 12h + 9 + h^2 - 2h + 1 = 1
$$

### Step 6: Combine like terms

Combining like terms results in:

$$
5h^2 - 14h + 10 = 1
$$

### Step 7: Rearrange the equation

Rearranging gives:

$$
5h^2 - 14h + 9 = 0
$$

### Step 8: Solve the quadratic equation

We can use the quadratic formula $$ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $$ a = 5, b = -14, c = 9 $$:

Calculating the discriminant:

$$
b^2 - 4ac = (-14)^2 - 4 \cdot 5 \cdot 9 = 196 - 180 = 16
$$

Now applying the quadratic formula:

$$
h = \frac{14 \pm \sqrt{16}}{2 \cdot 5} = \frac{14 \pm 4}{10}
$$

This gives us two possible values for $$ h $$:

1. $$ h = \frac{18}{10} = 1.8 $$
2. $$ h = \frac{10}{10} = 1 $$

### Final Answer

The possible values of $$ h $$ are:

$$
h = 1.8 \quad \text{or} \quad h = 1
$$
The possible values of $$ h $$ are 1.8 and 1.

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