If a , b, c be the vectors determined by sides BC, CA and AB of a triangle ABC and of magnitude a, b, c then are the following relations true or false :
(i) `a^(2) = b^(2) + c^(2) - 2bc cos A`
(ii) `a = b cos C + c cos B `
(iii) `|a xx b |= |b xx c| = |c xx a|`
(iv) `(sin A)/(a) = (sin B)/(b) = (sin C)/(c)`

Answer

To determine whether the given relations involving the vectors \( a, b, c \) of triangle \( ABC \) are true or false, we will analyze each statement one by one.

### Given:
- \( a, b, c \) are the vectors determined by the sides \( BC, CA, AB \) of triangle \( ABC \).
- Magnitudes of these vectors are denoted as \( |a|, |b|, |c| \).

### (i) \( a^2 = b^2 + c^2 - 2bc \cos A \)

**Step 1:** Recognize that this is the cosine rule, which relates the lengths of the sides of a triangle to the cosine of one of its angles.

**Step 2:** According to the cosine rule, for triangle \( ABC \):
\[ a^2 = b^2 + c^2 - 2bc \cos A \]
This is a standard result in triangle geometry.

**Conclusion:** This statement is **True**.

### (ii) \( a = b \cos C + c \cos B \)

**Step 1:** This statement refers to the projection of vectors.

**Step 2:** In triangle \( ABC \), the length of side \( a \) can be expressed as the sum of the projections of sides \( b \) and \( c \) onto the direction of side \( a \).

**Step 3:** Using the projection formula:
\[ a = b \cos C + c \cos B \]
This is indeed a valid expression for the length of side \( a \).

**Conclusion:** This statement is **True**.

### (iii) \( |a \times b| = |b \times c| = |c \times a| \)

**Step 1:** The magnitude of the cross product of two vectors gives the area of the parallelogram formed by those vectors.

**Step 2:** The area of triangle \( ABC \) can be expressed in terms of any two sides and the sine of the included angle.

**Step 3:** Therefore, the areas represented by \( |a \times b|, |b \times c|, |c \times a| \) are all equal to twice the area of triangle \( ABC \).

**Conclusion:** This statement is **True**.

### (iv) \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)

**Step 1:** This is known as the sine rule, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant for all three sides.

**Step 2:** Therefore, we can express:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
This implies that:
\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \]

**Conclusion:** This statement is **True**.

### Final Summary:
- (i) True
- (ii) True
- (iii) True
- (iv) True