https://file.aiquickdraw.com/mathbot/file/17301253981066g1qal3i.jpg

Let's analyze the vector **BC** based on the given image, where we need to:

1. Represent **BC** in column vector form.
2. Express **BC** in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \).
3. Calculate the magnitude (length) of vector **BC**.

### Solution:

From the graph, we observe:
- The coordinates of point \( B \) are \( (3, 2) \).
- The coordinates of point \( C \) are \( (7, 4) \).

### Step-by-Step Solution:

#### (a) Column Vector Form of **BC**

The column vector form for a vector **BC** from point \( B(x_1, y_1) \) to \( C(x_2, y_2) \) is:
\[
\mathbf{BC} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix}
\]
Using \( B(3, 2) \) and \( C(7, 4) \):
\[
\mathbf{BC} = \begin{pmatrix} 7 - 3 \\ 4 - 2 \end{pmatrix} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}
\]

#### (b) Unit Vector Form \( \mathbf{i} \) and \( \mathbf{j} \)

Expressing **BC** in terms of unit vectors \( \mathbf{i} \) (horizontal) and \( \mathbf{j} \) (vertical):
\[
\mathbf{BC} = 4\mathbf{i} + 2\mathbf{j}
\]

#### (c) Magnitude of Vector **BC**

The magnitude \( |\mathbf{BC}| \) of the vector **BC** is calculated by:
\[
|\mathbf{BC}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the values:
\[
|\mathbf{BC}| = \sqrt{(7 - 3)^2 + (4 - 2)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.47
\]

### Summary of Answers:
- **(a)** Column vector form: \( \begin{pmatrix} 4 \\ 2 \end{pmatrix} \)
- **(b)** Unit vector form: \( 4\mathbf{i} + 2\mathbf{j} \)
- **(c)** Magnitude of vector **BC**: \( 2\sqrt{5} \approx 4.47 \)

Would you like additional details on any part of the solution?

### Related Questions
1. How do you calculate the midpoint of a vector given its endpoints?
2. What is the direction angle of vector **BC** with respect to the x-axis?
3. How do you find the unit vector in the direction of vector **BC**?
4. How would the solution change if the vector pointed from \( C \) to \( B \) instead?
5. What is the general formula for converting between vector forms?

### Tip
When calculating vector magnitudes, remember to square each component, add them, and then take the square root for the final result.

BC adalah sebuah vektor dengan titik pangkal B dan titik ujung C. Tentukan: a. BC dalam bentuk vektor kolom, b. BC dalam bentuk i, j (vektor satuan), c. Panjang vektor BC.

Solve for vector BC in three forms: column vector, unit vector (i, j), and magnitude. Understand how to represent vectors in different forms using the coordinates of points B and C. Suitable for students in Grades 9-10.

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