Let 3 𝐢-𝐣+4 𝐤, j⃗-2 𝐤, 𝐢-3 𝐣+𝐤 be three vectors with tails at the origin. Then their heads deter

step 1 Hi everyone here it is given position vector A 3i cap minus j cap plus 4k cap V is 7 j cap minus

Let 3 𝐢-𝐣+4 𝐤, j⃗-2 𝐤, 𝐢-3 𝐣+𝐤 be three vectors with tails...

Let $3 \mathbf{i}-\mathbf{j}+4 \mathbf{k}, \vec{j}-2 \mathbf{k}, \mathbf{i}-3 \mathbf{j}+\mathbf{k}$ be three vectors with tails at the origin. Then their heads determine three points $A, B, C$ in space which form a triangle. Find vectors representing the sides $A B, B C, C A$ in that order and direction (for example, $A$ to $B$, not $B$ to $A$ ) and show that the sum of these vectors is zero.

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Key Concepts

Vector Subtraction

Vector subtraction is a fundamental operation where the difference between two position vectors gives the directed vector connecting the initial point of the first vector to the terminal point of the second vector. This operation is key when finding side vectors of a polygon, such as determining the sides of a triangle by subtracting coordinates of the endpoints appropriately.

Closed Polygon and Zero Vector Sum

The property that the sum of the side vectors of a triangle, when traversed in order, equals the zero vector arises because the sides form a closed loop. This is a manifestation of the head-to-tail principle in vector addition, which states that systematically adding vectors along a closed path yields a net displacement of zero.

Vector Addition

Vector addition is the process of combining vectors to determine a cumulative displacement. It plays a crucial role in verifying properties of geometric figures such as triangles, where the combined effect of moving along all the sides returns one to the starting point, thus summing to the zero vector.

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