This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for the vector problem. The given vectors are: a = 3i + 4j b = -6i + 8j c = -6i - 8j Step 1: Identify equal vectors. Two vectors are equal if their corresponding components are identical. Comparing the components: a = (3, 4) b = (-6, 8) c = (-6, -8) No two vectors have identical components. Therefore, no two vectors are equal. Step 2: Identify parallel vectors. Two vectors u and v are parallel if v = ku for some non-zero scalar k. This means their corresponding components must be proportional. Check a and b: Assume b = ka: -6i + 8j = k(3i + 4j) -6i + 8j = 3ki + 4kj Equating the components: -6 = 3k k = -2 8 = 4k k = 2 Since the value of k is not consistent (-2 ≠ 2), a and b are not parallel. Check a and c: Assume c = ka: -6i - 8j = k(3i + 4j) -6i - 8j = 3ki + 4kj Equating the components: -6 = 3k k = -2 -8 = 4k k = -2 Since k = -2 is consistent for both components, a and c are parallel. Check b and c: Assume c = kb: -6i - 8j = k(-6i + 8j) -6i - 8j = -6ki + 8kj Equating the components: -6 = -6k k = 1 -8 = 8k k = -1 Since the value of k is not consistent (1 ≠ -1), b and c are not parallel. Therefore, the parallel vectors are a and c. Step 3: Identify opposite vectors. Strictly, two vectors u and v are opposite if v = -u. This means they have the same magnitude and point in exactly opposite directions. First, calculate the magnitudes of the vectors: |a| = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 |b| = sqrt((-6)^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 |c| = sqrt((-6)^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10 For vectors to be opposite, their magnitudes must be equal. Only b and c have equal magnitudes. Let's check if c = -b: -b = -(-6i + 8j) = 6i - 8j Since c = -6i - 8j and -b = 6i - 8j, they are not equal. Thus, b and c are not opposite. Under the strict definition, no two vectors are opposite. However, "opposite" can also be interpreted as "parallel and pointing in the opposite direction" (i.e., v = ku where k < 0). From Step 2, we found that a and c are parallel, with c = -2a. Since k = -2 (which is negative), a and c are parallel and point in opposite directions. Therefore, the vectors that are parallel and point in opposite directions are a and c. Last free one today — make it count tomorrow, or type /upgrade for unlimited.