Question 1 (30 Marks) (a) Given vectors P=5i+2j-4k and Q=2i-3j+5k . Evaluate the following (i) (5P+2Q)-(2P-3Q) (ii) Determine the magnitude of (3P+2Q)-(2P-3Q)

(i) To evaluate $(5P+2Q)-(2P-3Q)$, we first need to find the expressions for $5P+2Q$ and $2P-3Q$.<br /><br />$5P+2Q = 5(5i+2j-4k) + 2(2i-3j+5k$= 25i + 10j - 20k + 4i - 6j + 10k$<br />$= 29i + 4j - 10k$<br /><br />$2P-3Q = 2(5i+2j-4k) - 3(2i-3j+5k)$<br />$= 10i + 4j - 8k - 6i + 9j - 15k$<br />$= 4i + 13j - 23k$<br /><br />Now, we can find $(5P+2Q)-(2P-3Q)$:<br />$(5P+2Q)-(2P-3Q) = (29i + 4j - 10k) - (4i + 13j - 23k)$<br />$= 29i + 4j - 10k - 4i - 13j + 23k$<br />$= 25i - 9j + 13k$<br /><br />(ii) To determine the magnitude of $(3P+2Q)-(2P-3Q)$, we first need to find the expression for $(3P+2Q)-(2P-3Q)$.<br /><br />$(3P+2Q)-(2P-3Q) = 3(5i+2j-4k) + 2(2i-3j+5k) - (4i + 13j - 23k)$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + $<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13j + 23k$<br />$= 15i + 6j - 12k + 4i - 6j + 10k - 4i - 13